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Graphing Linear Equations
8th Grade
www.njctl.org
2015-01-26
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click on the topic to go to that section
Table of Contents· Vocabulary Review
· Defining Slope on the Coordinate Plane· Tables and Slope
· Tables
· Slope Formula
· Slope & y-intercept
· Slope Intercept Form· Rate of Change· Proportional Relationships and Graphing· Slope and Similar Triangles· Parallel and Perpendicular Lines· Solve Systems by Graphing· Solve Systems by Substitution· Solve Systems by Elimination· Choosing Your Strategy· Writing Systems to Model Situations· Glossary
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Links to PARCC sample questions
Calculator #6
Calculator #7
Calculator #9
Non-Calculator #3 Calculator #4
Non-Calculator #11
Non-Calculator #16
Non-Calculator #7
Calculator #8
Calculator #12
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Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number.
How many thirds are in 1 whole?
How many fifths are in 1 whole?
How many ninths are in 1 whole?
Vocabulary words are identified with a dotted underline.
The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.
(Click on the dotted underline.)
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Back to
Instruction
FactorA whole number that can divide into another number with no remainder.
15 3 5
3 is a factor of 153 x 5 = 15
3 and 5 are factors of 15
1635 .1R
3 is not a factor of 16
A whole number that multiplies with another number to make a third number.
The charts have 4 parts.
Vocab Word1
Its meaning 2
Examples/ Counterexamples
3Link to return to the instructional page.
4
(As it is used in the
lesson.)
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y-axis: vertical number line that extends indefinitely in both directions from zero. (Up- positive Down- negative)
x-axis: horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left- negative)
Origin: the point where zero on the x-axis intersects zero on the y-axis. The coordinates of the origin are (0,0).
II I
III IV
Vocabulary ReviewCoordinate Plane: the two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. Also known as a coordinate graph and the Cartesian plane.
Quadrant: any of the four regions created when the x-axis intersects the y-axis. They are usually numbered with Roman numerals.
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To graph an ordered pair, such as ( 4, 8), you start at the origin (0, 0)and then go left or right on the x-axis depending on the first number and then up or down from there parallel to the y-axis.
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(4,8)
So to graph (4,8), we would go 4 to the right and up 8 from there.
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Linear Equation:
Any equation whose graph is represented by a straight line.
One way to check this is to create a table of values.
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Geometry Theorem:
Through any two points in a plane there can be drawn only one line.
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Given y=3x+2, we want to graph our equation to show all of the ordered pairs that make it true.
So according to this theorem from Geometry, we need to find 2 points.
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One way is to create a table of values.
Let's consider the equation y= 3x + 2.
We need to find pairs of x and y numbers that make equation true.
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Let's find some values for y=3x+2.
Pick values for x and plug them into the equation,then solve for y.
x 3(x)+2 y (x,y)
0 3(0)+2 2 (0,2) 2 3(2)+2 8 (2,8)
-3 3(-3)+2 -7 (-3,-7)
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x 3(x)+2 y (x,y)
0 3(0)+2 2 (0,2) 2 3(2)+2 8 (2,8)
-3 3(-3)+2 -7 (-3,-7)
Now let's graph those points we just found.
Notice anything about the points we just graphed?
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That's right! The points we graphed form a line.
The theorem says we only needed 2 points, so why did we graph 3 points?
The third point serves as a check.
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Graph y = 2x+4
x 2x+4 y (x,y)0 2(0)+4 4 (0,4)3 2(3)+4 10 (3,10)-1 2(-1)+4 2 (-1,2)
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Now graph your pointsand draw the line.
y
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Click for graph
x 2x+4 y (x,y)
click for table
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x -2(x)+1 y (x,y)0 -2(0)+1 1 (0,1)3 -2(3)+1 -5 (3,-5)-1 -2(-1)+1 3 (-1,3)
Graph y = -2x+1
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Now graph your pointsand draw the line.
y
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x -2(x)+1 y (x,y)
click for table
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Graph y = ¾x-3
x ¾(x )-3 y (x,y)0 ¾(0)-3 -3 (0,-3)4 ¾(4)-3 0 (4,0)-4 ¾(-4)-3 -6 (-4,-6)
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Now graph your pointsand draw the line.
x ¾(x)-3 y (x,y)click for table
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Recall that in the previous examplethat even though the number in frontof x was a fraction, our answers wereintegers.
Why? Discuss at your table.
x ¾(x )-3 y (x,y)0 ¾(0)-3 -3 (0,-3)4 ¾(4)-3 0 (4,0)-4 ¾(-4)-3 -6 (-4,-6)
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11 A solution is 20% bleach.
Create a graph that represents all possible combinations of the number of liters of bleach, contained in the number of liters of the solution.
To graph a line, pick two points on the coordinate plane. A line will be drawn through the points.
Students type their answers here
From PARCC sample test
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You only need a few facts about a line to completely describe it:
· Its y-intercept (where it crosses the y-axis)
"b"
· Its slope (how much it rises or falls)
"m"
· y = mx + b
The Equation of a Line
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Imagine trying to tell a person how to draw a line on the Cartesian
Plane.
Consider this graph of the Cartesian Plane, also called a Coordinate Plane or
XY-Plane.
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The y-intercept ("b")of a line is the point where the line intercepts the y-axis.
In this case, the y-intercept of the line is +4.
The y-intercept
This is the ordered pair (0,4).
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12 What is the y-intercept of this line?
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13 What is the y-intercept of this line?
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14 What is the y-intercept of this line?
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15 What is the x-intercept of this line?
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16 What is the x-intercept of this line?
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17 What is the x-intercept of this line?
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18 The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are
A (0,2)B (0,6)C (0,18) D (6,0)
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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"Steepness" and "Position" of a Line
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An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept.
Examples of lines with a y-intercept of ____ are shown on this graph. What's the difference between them (other than their color)?
Consider this...
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The lines all have a different slope.
Slope is the steepness of a line.
Compare the steepness of the lines on the right.
Slope can also be thought of as the rate of change.
The Slope of a Line
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The red line has a positive slope, since the line rises from left to the right.
The Slope of a Linerun
rise
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The orange line has a negative slope, since the line falls down from left to the right.
The Slope of a Line
riserun
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The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis.
The Slope of a Line
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The Slope of a Line
The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis.
rise0
= undefined
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While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line.
Measuring the Slope of a Line
rise
run
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The slope of the line is just the ratio of its rise over its run.
The symbol for slope is "m".
So the formula for slope is:
Measuring the Slope of a Line
rise
run
slope = riserun
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The slope is the same anywhere on a line, so it can be measured anywhere on the line.
Measuring the Slope of a Line
rise
run
slope = riserun
Keep in mind the direction:· Up (+) Down (-)· Right (+) Left (-)
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For instance, in this case we measure the slope by using a run from x = 0 to x = +6: a run of 6.
During that run, the line rises from y = 0 to y = 8: a rise of 8.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 86
m = 43
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But we get the same result with a run from x = 0 to x = +3: a run of 3.
During that run, the line rises from y = 0 to y = 4: a rise of 4.
Measuring the Slope of a Line
riserun
slope = riserun
m = 43
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But we can also start at x = 3 and run to x = 6 : a run of 3.
During that run, the line rises from y = 3 to y = 7: a rise of 4.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 43
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But we can also start at x = -6 and run to x = 0: a run of 6.
During that run, the line rises from y = -8 to y = 0: a rise of 8.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 86
m = 43
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How is the slope different on this coordinate plane?
The line rises 8, however the run goes left 6(negative). Therefore, it is said to have a negative slope
Measuring the Slope of a Line
rise
run
slope = riserun
m = 8-6
m = -4 3
*most often the negative sign is placed in the numerator
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x y
-3 -1
0 5
3 11
How can slope and the y-intercept be found within the table?
· Look for the change in the y-values· Look for the change in the x-values· Write as a ratio (simpified) - this will be the "slope"· Determine the corresponding y-value to the x-value of 0 - this will be the "y-intercept"
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x y
-3 -1
0 5
3 11
+6
+6
+3
+3
63
= 2 is the slope 5 is the y-intercept
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x y
5 -5
0 -4
-5 -3
-4 is the y-intercept is the slope
Determine the slope and y-intercept from this table.
click to reveal answer
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Slope is "the rise over the run"of a line.
This idea of rise over run of a line on a graph is how we were able to determine the slope of a line.
But slope can be found in other ways than looking at a graph.
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Slope is the ratio of change in y (rise) divided by the change in x(run). slope= =
A line has a constant ratio of change:A constant increase
A constant decrease
No change, just constant
Or undefined slope
riserun
change in ychange in x
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Another Application of the Definition of Slope
Slope of a line is meant to measure how fast it is climbing or descending.
A road might rise 1 foot for every 10 feet of horizontal distance.
10 feet
1 foot
The ratio, 1/10, which is called slope, is a measure of the steepness of the hill. Engineers call this use of slope grade.
What do you think a grade of 4% means?
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Slope of 3/20
20 feet
3 feet
3 feet
7 feet
slope of -3/7(The grade of this hill is
3/20 = .15= 15%)
(The grade of this hill is 3/7 = .43= 43%)
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so we will define the slope of a line as:
slope = vertical change between two point on the line
horizontal change between two point on the line
(Rise)
(Run)
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Suppose point P = (x1, y1) and Q = (x2, y2) are on the line whose slope we want to find.
Q(x2,y2)
P(x1,y1)
x
y
(x2,y1)Horizontal Change(x2-x1)
VerticalChange(y2-y1)
The slope of line PQ=(y2-y1)(x2-x1)
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The vertical change between P and Q = y2 - y1
The horizontal change = x2 - x1
y2 - y1
x2 - x1slope =
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Once you have identified the slope and y-intercept in an equation, it is easy to graph it!
To graph y = 3x + 5...follow these steps:· Plot the y-intercept, in this case (0, 5)· Use the simplified rise over run to plot the next point - in
this case, from (0, 5) go UP 3 units and RIGHT 1 unit to plot the next point. Connect the points.
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Try this...graph y = -2x - 3
· Start at the y-intercept - plot it.· From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot -2?· Connect the points.
click to reveal
Did you have different points plotted? Does it make a difference?
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Try this...graph 4y = x + 12 (is this in y=mx + b form??)
· Start at the y-intercept - plot it.· From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it?· Connect the points.
Did you have different points plotted? Does it make a difference?
click to reveal
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Try this...graph 5x + y = -4 (is this in y=mx + b form??)
· Start at the y-intercept - plot it.· From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it?· Connect the points.
Did you have different points plotted? Does it make a difference?
click to reveal
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Position of a Line
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What are the similarities and differences between the lines below?
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h(x)=x+6
q(x)=x+2
r(x)=x-1 s(x)=x-5
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The lines were in the form of y = mx+b.
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So it is the b in y = mx + b that is responsible for the position of the line.
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h(x)=x+6
q(x)=x+2
r(x)=x-1 s(x)=x-5
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What determines slope?
Examine the following equations:
y = 2x + 1 y = 3x + 1
y = -1/2 x + 1 y = -x + 1
What do the equations have in common?
What is different?
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y=-7x+1
y=-1/2x+1
y=1
y=x+1y=-3x+1
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Any equation of the form y = mx + b gives a line where
b is the y intercept
m is the slope
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Click for an interactive web site to see how the position of the line changes as you change
the slope and the y-intercept.
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Slope formula can be used to find the constant of change in a "real world" problem.
When traveling on the highway, drivers will set the cruise control and travel at a constant speed this means that the distance traveled is a constant increase.
The graph at the right represents such a trip. The car passed mile-marker 60 at 1 hour and mile-marker 180 at 3 hours. Find the slope of the line and what it represents.
m= = =
So the slope of the line is 60 and the rate of change of the car is 60 miles per hour.
180 miles-60 miles 3 hours-1 hours
120 miles 2 hours
60 miles hour
Time(hours)
Distance(miles)
(1,60)
(3,180)
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If a car passes mile-marker 100 in 2 hours and mile-marker 200 in 4 hours, how many miles per hour is the car traveling?
The information above gives us the ordered pairs (2,100) and (4,200). Now find the rate of change.
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61 Two different proportional relationships are represented by the equation and the table. Proportion A Proportion B
The rate of change in Proportion A is ______ ______ then the rate of change to Proportion B.
A 1.5
B 2.5
C 25.5
D 43.5
E more
F less
From PARCC sample test
y = 9x
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62 A pool cleaning service drained a full pool. The following table shows the number of hours it drained and the amount of water remaining in the pool at that time.
Part A
Plot the points that show the relationship between the number of hours elapsed and the number of gallons of water left in the pool.
Select a place on the grid to plot each point. (Grid on next slide.)
Students type their answers here
From PARCC sample test
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63 Part B (continued from previous question)
The data suggests a linear relationship between the number of hours the pool had been draining and the number of gallons of water remaining in the pool. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship.
A The number of gallons of water in the pool after 1 hour.
B The number of hours it took to drain 1 gallon of water.
C The number of gallons drained each hour.
D The number of gallons of water in the pool when it is full.
From PARCC sample test
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64 Part C (continued from previous question)
What does the y-intercept of the linear function repressent in the context of this relationship?
A The number of gallons in the pool after 1 hour.
B The number of hours it took to drain 1 gallon of water.
C The number of gallons drained each hour.
D The number of gallons of water in the pool when it is full.
From PARCC sample test
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65 Part D (continued from previous question)
Which equation describes this relationship between the time elapsed and the number of gallons of water remaining in the pool?
A y = -600x + 15,000
B y = -600x + 13,2000
C y = -1,200x + 13,200
D y = -1,200x + 15,000
From PARCC sample test
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66 Eric planted a seedling in his garden and recorded its height each week. The equation shown can be used to estimate the height h, in inches, of the seedling after w, weeks since Eric planted the seedling.
Part A: What does the slope of the graph of the equation represent?
A The height in inches, of the seedling after w weeks.
B The height in inches, of the seedling when Eric planted it.
C The increases of height in inches, of the seedling each week.
D The total increase in the height in inches, of the seedling after w weeks.
From PARCC sample test
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67 Part B (continued from previous question)
The equation estimates the height of the
seedlings to be 8.25 inches after how many weeks?
From PARCC sample test
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Pavers are being set around a birdbath. The figures below show the first three designs of the pattern.
Using tiles, build the first five designs that follow the pattern above. Record your results in a table.
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Design number 1 2 3 4 5
Number of pavers 4 8 12 16 20
Graph the data from the table on a coordinate plane. What will you label the x-axis?What will you label the y-axis?
Do the coordinate pairs in your table represent a proportional relationship?
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How many tiles will you need for the "n-th" design? Write an equation that would represent the total number of tiles required for any design level.
Suppose the birdbath was replaced with two tiles...how would this change the pattern? How would this change the equation?
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t=4n
t=4n+2
Num
ber o
f tile
s
Design Level0 1 2 3 4 5 6
0
5
10
15
Graph both equations on the same coordinate plane. Discuss the similarities and differences in the graphs...
Num
ber o
f tile
s
Design Level0 1 2 3 4 5 6
0
5
10
15
Click for answer
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Congruent triangles have the same shape and same size. Using the line as the hypotenuse, draw congruent right triangles. How do you know they are congruent?
click to reveal example
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2
22
2
44
44
The vertical rise is the same as well as the horizontal run. The simplified ratio is the same as the absolute value of the slope.
24
12
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Similar triangles have the same shape, however, they are not the same size. The corresponding sides are proportionate.
4
6
234
623
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41
82
Sketch two similar right triangles on the line below. Write the ratios to prove they are proportionate.
click to reveal example
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75 Line t and ΔECA and ΔFDB are shown on the coordinate grid. Which statements are true? Select all that apply.
A The slope of AC is equal to the slope of BC.
B The slope of AC is equal to the slope of BD.
C The slope of AC is equal to the slope of line t.
D The slope of line t is equal to
E The slope of line t is equal to
F The slope of line t is equal to
y
t
x
From PARCC sample test
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Time (hr.)Dis tance (mi.)
from home
0 0
3 210
5 350
Time (hr.)Dis tance (mi.)
from home
0 10
3 220
5 360
Family AFamily Z
Slope (m) = 70y-intercept (b) = 0equation y = 70x
Slope (m) = 70y-intercept (b) = 10equation y = 70x + 10
Complete the items below each table.(Click boxes to reveal answers)
If this data from both tables were graphed on the same coordinate plane, what would you notice?
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The lines at the right are parallel lines. Notice that their slopes are all the same.
Parallel lines all have the slopes because if they change at different rates eventually they would intersect.
This also works for vertical and horizontal lines.
2
4
6
8
10
-2
-4
-6
-8
-10
2 4 6 8 10-2-4-6-8-10 0
h(x)=x+6
q(x)=x+2r(x)=x-1 s(x)=x-5
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In the diagram the 2 lines form a right angle, when this happenslines are said to perpendicular.
Look at their slopes. This time theyare not the same instead they areopposite reciprocals
h(x)=-3x-11
g(x)=1/3x-2
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A) y=4x-2 is perpendicular to
B) y=-1/5x+1 is perpendicular to
C) y-2=-1/4(x-3) is perpendicular to
D) 5x-y=8 is perpendicular to
E) y=1/6x is perpendicular to
F) y-9=-5(x-.4) is perpendicular to
G) y=-6(x+2) is perpendicular to
Perpendicular Equation Bank
y=1/6x-6
y=-1/4x-3y=4x+1
6x+y=10
1/5y=x-2
y=-1/5x+9
y=1/5x
(Drag the equationto complete the
statement.)
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The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines.
Why? Discuss with your table.
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Some vocabulary...
The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection.
A "system" is two or more linear equations.
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Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend?
Consider this...
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Time (min.)
Friend's distance from
your start (blocks)
Your distance from your start
(blocks)
0 5 0
1 6 2
2 7 4
3 8 6
4 9 8
5 10 10
First, make a table to represent the problem.
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Next, plot the points on a graph.
Time (min.)
Blo
cks
05
20
15
10
1510
5
0
Time (min.
)
Friend's distance from your
start (blocks)
Your distance from your
start(blocks)
0 5 0
1 6 2
2 7 4
3 8 6
4 9 8
5 10 10
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The point where they intersect is the solution to the system.
Time (min.)
Blo
cks
05
20
15
10
1510
5
0
(5,10) is the solution. In the context of the problem this means after 5 minutes, you will meet your friend
at block 10.
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Solve the system of equations graphically.
y = 2x -3y = x - 1
Solu
tion
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Solve the system of equations graphically.
2x + y = 3x - 2y = 4
Solu
tion
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Solve the system of equations graphically.
3x + y = 11x - 2y = 6
Solu
tion
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Solve using graphingy = 4x+6y = -3x-1move
Write the equation forthe green dashed line
Write the equation forthe blue solid line
What is this pointof intersection?(move the hand!) (-1, 2)
move
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( , )-1 2y = 4x+6y = -3x-1
Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines.
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y = 2x + 3
Solve by Graphingy = -4x - 3
Solu
tion
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y= x - 4y= -3x + 4
Solve by Graphing
Solu
tion
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What's the problem here? y= 2x - 4y= 2x + 4
Parallel lines do not intersect!
Therefore there is no solution.
No ordered pair that will work in BOTH equations
( )click to reveal
click to reveal
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2y = -4x + 102 2 y = -2x + 5
2x + y = 5 -2x -2x y = -2x + 5
Solve by GraphingFirst - transform the equations into y = mx + b
form (slope-intercept form)
Now graph the two transformed lines.
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2y = 10 -4x becomesy = -2x + 5
2x + y = 5 becomesy = -2x + 5
What's the problem?
The equations
transform to the same
line.
So we have infinitely
many solutions.
click to reveal click to reveal
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85 Solve the system by graphing.y = -x + 4y = 2x +1
A (3,1)
B (1,3)
C (-1,3)
D no solution
Click for multiple choice answers.
Solu
tion
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86 Solve the system by graphing.y = 0.5x - 1y = -0.5x -1
A (0,-1)
B (0,0)
C infinitely many
D no solution
Solu
tion
Click for multiple choice answers.
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87 Solve the system by graphing.2x + y = 3x - 2y = 4
A (2,4)
B (0.4, 2.2)
C (2, -1)
D no solution
Solu
tion
Click for multiple choice answers.
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88 Solve the system by graphing.y = 3x + 3y = 3x - 3
A (0,0)
B (3,3)
C infinitely many
D no solution
Solu
tion
Click for multiple choice answers.
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89 Solve the system by graphing.y = 3x + 44y = 12x + 16
A (3,4)
B (-3,-4)
C infinitely many
D no solution
Click for multiple choice answers.
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90 On the accompanying set of axes, graph and label the following lines: y=5 x = - 4 y = x+5
Calculate the area, in square units, of the triangle formed by the three points of intersection.
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
tion
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91 The equation of the line s is
The equation of the line t is
The equations of the lines s and t form a system of equations.
The solution of equations is located at Point P. Students type their answers here
From PARCC sample test
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92 The table shows two systems of linear equations. Indicate whether each system of the equations has no solution, one solution or infinitely many solutions by selecting the correct cell in the table. Select one cell per column.
Students type their answers here
From PARCC sample test
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Solve the system of equations graphically.
y = x + 6.1y = -2x - 1.4
NO
TE
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Graphing can be inefficient or approximate.
Another way to solve a system is to use substitution.
Substitution allows you to create a one variable equation.
Substitution Explanation
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Solve the system using substitution. Why was it difficult to solve this system by graphing?
y = x + 6.1y = -2x - 1.4
y = -2x - 1.4 -start with one equationx + 6.1 = -2x - 1.4 -substitute x + 6.1 for y in equation+2x -6.1 +2x - 6.1 3x = -7.5 -solve for x x = -2.5
Substitute -2.5 for x in either equation and solve for y. y = x + 6.1 y = ( -2.5) + 6.1 y = 3.6
Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6)
CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x - 1.43.6 = -2(-2.5) - 1.43.6 = 5 - 1.43.6 = 3.6?
?
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+ 3x = 21-3 y
y = -2x +14
Solve the system using substitution.
( )
(*Note: Equations can be moved on the page to show substitution into the y of the second equation.)
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= -y - 3x
x = -5y - 39
Solve the system using substitution.
( )
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Examine each system of equations.Which variable would you choose to substitute?Why?
y = 4x - 9.6y = -2x + 9
y = -3x7x - y = 42
y = 4x + 1x = 4y + 1
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93 Examine the system of equations. Which variable would you substitute?
2x + y = 52y = 10 - 4x
A x B y
Solu
tion
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94 Examine the system of equations. Which variable would you substitute?
2y - 8 = xy + 2x = 4
A x B y
Solu
tion
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95 Examine the system of equations. Which variable would you substitute?
x - y = 202x + 3y = 0
A x B y
Solu
tion
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Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example:
The system: Is equivalent to:3x -y = 5 y = 3x -52x + 5y = -8 2x + 5y = -8
Using substitution you now have: 2x + 5(3x-5) = -8 -solve for x2x + 15x - 25 = -8 -distribute the 5 17x - 25 = -8 -combine x's 17x = 17 -at 25 to both sides x = 1 - divide by 17
Substitute x = 1 into one of the equations.2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2
The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5 2x + 5y = -83(1) - (-2) = 5 2(1) + 5(-2) = -8 3 + 2 = 5 2 - 10 = -8 -8 = -8
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Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip?
Let v = the number of vansand c = the number of cars
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Set up the system: Drivers: v + c = 4 People: 6v + 4c = 22Solve the system by substitution. v + c = 4 -solve the first equation for v. v = -c + 4 -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22 second equation -6c + 24 + 4c = 22 -solve for c -2c + 24 = 22 -2c = -2 c = 1
v + c = 4 v + 1 = 4 -substitute for c in the 1st equation v = 3 -solve for v
Since c = 1 and v = 3, they should use 1 car and 3 vans.
Check the solution in the equations: v + c = 4 6v + 4c = 22 3 + 1 = 4 6(3) + 4(1) = 22 4 = 4 18 + 4 = 22 22 = 22
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Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10
x + y = 6 -solve the first equation for x x = 6 - y5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30 - 5y + 5y = 10 -solve for y 30 = 10 -FALSE!
Since 30 = 10 is a false statement, the system has no solution.
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Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6
x + 4y = -3 - solve the first equation for x x = -3 - 4y2(-3 - 4y) + 8y = -6 - sub. -3 - 4y for x in 2nd equation -6 - 8y + 8y = -6 - solve for y -6 = -6 - TRUE! - there are infinitely many solutions
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How can you quickly decide the number of solutions a system has?
1 Solution Different slopes
No SolutionSame slope; different y-intercept (Parallel Lines)
Infinitely Many Same slope; same y-intercept (Same Line)
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96 3x - y = -2 y = 3x + 2
A 1 solution
B no solution
C infinitely many solutions
Solu
tion
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97 3x + 3y = 8 y = x
A 1 solution
B no solution
C infinitely many solutions
1 3
Solu
tion
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98 y = 4x 2x - 0.5y = 0
A 1 solution
B no solution
C infinitely many solutions
Solu
tion
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99 3x + y = 5 6x + 2y = 1
A 1 solution
B no solution
C infinitely many solutions
Solu
tion
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100 y = 2x - 7 y = 3x + 8
A 1 solution
B no solution
C infinitely many solutions
Solu
tion
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101 Solve each system by substitution.y = x - 3y = -x + 5
A (4,9)
B (-4,-9)
C (4,1)
D (1,4)
Solu
tion
Click for multiple choice answers.
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102 Solve each system by substitution.y = x - 6y = -4
A (-10,-4)
B (-4,2)
C (2,-4)
D (10,4)
Solu
tion
Click for multiple choice answers.
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103 Solve each system by substitution.y + 2x = -14y = 2x + 18
A (1,20)
B (1,18)
C (8,-2)
D (-8,2)
Solu
tion
Click for multiple choice answers.
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104 Solve each system by substitution.4x = -5y + 50x = 2y - 7
A (6,6.5)
B (5,6)
C (4,5)
D (6,5)
Solu
tion
Click for multiple choice answers.
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105 Solve each system by substitution.y = -3x + 23-y + 4x = 19
A (6,5)
B (-7,5)
C (42,-103)
D (6,-5)
Solu
tion
Click for multiple choice answers.
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When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination.
You can add or subtract the equations to eliminate a variable.
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How do you decide which variable to eliminate?
First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable.
Second, look for which coefficients have a simple least common multiple. Eliminate that variable.
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If the variables have the same coefficient, you can subtract the two equations to eliminate the variable.
If the variables have opposite coefficients, you add the two equations to eliminate the variable.
Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient.
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5x + y = 44-4x - y = -34
Solve by Elimination - Click on the terms to eliminate and they will disappear, then add
the two equations together.
)(
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3x + y = 15-3x -3y = -21
Solve by Elimination - Click on the terms and they will disappear then add the two
equations together.
( )
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5x + y = 17-2x + y = -4
Solve by Elimination - There are 2 ways to complete this problem. See both examples.
Mul
tiplic
atio
n by
-1
Sub
tract
ion
5x + y = 17-2x + y = -4
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Solve the system by elimination.
4x + 3y = 16 2x - 3y = 8
Pul
lP
ull
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106 Solve each system by elimination.x + y = 6x - y = 4
A (5,1)
B (-5,-1)
C (1,5)
D no solution
Solu
tion
Click for multiple choice answers.
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107 Solve each system by elimination.2x + y = -52x - y = -3
A (-2,1)
B (-1,-2)
C (-2,-1)
D infinitely many
Solu
tion
Click for multiple choice answers.
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108 Solve each system by elimination.2x + y = -63x + y = -10
A (4,2)
B (3,5)
C (2,4)
D (-4,2)
Solu
tion
Click for multiple choice answers.
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109 Solve each system by elimination.4x - y = 5x - y = -7
A no solution
B (4,11)
C (-4,-11)
D (11,-4)
Solu
tion
Click for multiple choice answers.
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110 Solve each system by elimination.3x + 6y = 48-5x + 6y = 32
A (2,-7)
B (7,2)
C (2,7)
D infinitely many
Solu
tion
Click for multiple choice answers.
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Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations.
When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations.
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Examine each system of equations.Which variable would you choose to eliminate?What do you need to multiply each equation by?
2x + 5y = -1 x + 2y = 0
3x + 8y = 815x - 6y = -39
3x + 6y = 62x - 3y = 4
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In order to eliminate the y, you need to multiply first.
3x + 4y = -10 5x - 2y = 18
Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36
Now solve by adding the equations together. 3x + 4y = -10 10x - 4y = 36 13x = 26 x = 2
Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = -10 6 + 4y = -10 4y = -16 y = -4
So (2,-4) is the solution.
Check: 3x + 4y = -10 5x - 2y = 183(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18
+
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Now solve the same system by eliminating x. What do you multiply the two equations by?
3x + 4y = -10 5x - 2y = 18
Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 3(5x - 2y = 18) 15x + 20y = -50 15x - 6y = 54
Now solve by subtracting the equations. 15x + 20y = -50 15x - 6y = 54 26y = -104 y = -4
Solve for x, by substituting y = -4 into one of the equations. 3x + 4y = -10 3x + 4(-4) = -10 3x + -16 = -10 3x = 6 x = 2
So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 183(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18
-
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111 Which variable can you eliminate with the least amount of work?
A xB y
9x + 6y = 15-4x + y = 3
Solu
tion
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112 Which variable can you eliminate with the least amount of work?
A xB y
3x - 7y = -2-6x + 15y = 9 So
lutio
n
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113 Which variable can you eliminate with the least amount of work?
A xB y
x - 3y = -72x + 6y = 34
Solu
tion
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114 What will you multiply the first equation by in order to solve this system using elimination?
2x + 5y = 203x - 10y = 37
Now solve it....
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3x + 2y = -19x - 12y = 19
Now solve it....
115 What will you multiply the first equation by in order to solve this system using elimination?
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x + 3y = 43x + 4y = 2
Now solve it....
116 What will you multiply the first equation by in order to solve this system using elimination?
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Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1.
Ticket sales were $470.
Let a = adults s = students
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Set up the system:
number of tickets sold: a + s = 292 money collected: 3a + s = 470
First eliminate one variable. a + s = 292 - in both equations s has the same - (3a + s = 470) coefficient so you subtract the 2 -2a+ 0 = -178 equations in order to eliminate it. a = 89 -solve for a
Then, find the value of the eliminated variable. a + s = 29289 + s = 292 -substitute 89 for a in 1st equation s = 203 -solve for s
There were 89 adult tickets and 203 student tickets sold.
(89, 203)
Check: a + s = 292 3a + s = 47089 + 203 = 292 3(89) + 203 = 470 292 = 292 267 + 203 = 470 470 = 470
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117 A piece of glass with an initial temperature of 99 º F is cooled at a rate of 3.5 º F/min. At the same time, a piece of copper with an initial temperature of 0 º F is heated at a rate of 2.5º F/min. Let m = the number of minutes and t = the temperature in F. Which system models the given information?
A B Ct = 99 + 3.5mt = 0 + 2.5m
t = 99 - 3.5mt = 0 + 2.5m
t = 99 + 3.5mt = 0 - 2.5m
Solu
tion
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118 Which method would you use to solve the system?
A graphingB substitutionC elimination
t = 99 - 3.5mt = 0 + 2.5m
Now solve it...m = 16.5 t = 41.25
This means that in 16.5 minutes, the temperatures will both be 41.25º C.click for answer
click for equations
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119 What method would you choose to solve the system?
A graphingB substitution
C elimination
4s - 3t = 8t = -2s -1
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D (-2, )
120 Now solve the system!
A ( , -2) 4s - 3t = 8t = -2s -1
1 2
B ( , 2)
1 2
C (2 , -2)
1 2
Click for multiple choice answers.
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121 What method would you choose to solve the system?
A graphingB substitutionC elimination
y = 3x - 1y = 4x
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122 Now solve it!
A (1, 4)
B (-4, -1)
C (-1, 4)
y = 3x - 1y = 4x
D (-1, -4)
Click for multiple choice answers.
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123 What method would you choose to solve the system?
A graphingB substitutionC elimination
3m - 4n = 13m - 2n = -1
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124 Now solve it!
A (-2, -1)
B (-1, -1)
C (-1, 1)
3m - 4n = 13m - 2n = -1
D (1, 1)
Click for multiple choice answers.
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125 What method would you choose to solve the system?
A graphing
B substitution
C elimination
y = -2xy = -0.5x + 3
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126 Now solve it!
A (-6, 12)
B (2, -4)
y = -2xy = -0.5x + 3
C (-2, 4)
D (1, -2)
Click for multiple choice answers.
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127 What method would you choose to solve the system?
A graphing
B substitution
C elimination
2x - y = 4x + 3y = 16
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128 Now solve it!
A (6, 5)
B (-4, 7)
C (-4, 4)
2x - y = 4x + 3y = 16
D (4, 4)
Click for multiple choice answers.
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129 What method would you choose to solve the system?
A graphing
B substitutionC elimination
u = 4v3u - 3v = 7
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130 Now solve it!
A ( , )B ( , )
C (28, 7)
u = 4v3u - 3v = 7
D (7, ) 7 4
28 9
28 9
7 9
7 9
Click for multiple choice answers.
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131 Choose a strategy and then answer the question.What is the value of the y-coordinate of the solution to the system of equations x − 2y = 1 and x + 4y = 7?
A 1B -1C 3D 4
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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132 A system of equations is shown.
What is the solution (x,y) of the system of equations?
x = _____ y = ____
Students type their answers here
From PARCC sample test
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133 Two lines are graphed on the same coordinate plane. The lines only intersect at the point (3,6). Which of these systems of linear equations could represent the two lines?
Select all that apply.
A
B
C
D
E
From PARCC sample test
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A group of 148 people is spending five days at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,410 pounds of food was ordered.
Part A: Write an equation or a system of equations that describes the above situation and define your variables.
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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Part B: Using your work from part A, find:
(1) the total number of adults in the group
(2) the total number of children in the group
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Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha’s bill was $6.00, and Rachel’s bill was $5.25. What was the price of one slice of pizza? What was the price of one cola?
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From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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Sharu has $2.35 in nickels and dimes. If he has a total of thirty-two coins, how many of each coin does he have?
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From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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Ben had twice as many nickels as dimes. Altogether, Ben had $4.20.How many nickels and how many dimes did Ben have?
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
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134 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9
Set up a system and solve. How many packages of cards were sold?
You will answer how many packages of gift
wrap in the next question.
Solu
tion
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135 Your class receives $1105 for selling 205 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9
Set up a system and solve. How many packages of gift wrap were sold?
Solu
tion
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136 The sum of two numbers is 47, and their difference is 15. What is the larger number?
A 16B 31C 32D 36
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
tion
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137 Ramon rented a sprayer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $90. On his second job, he used the sprayer for 4 hours and the generator for 8 hours at a total cost of $100. What was the hourly cost for the sprayer?
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
tion
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138 What is true of the graphs of the two lines 3y - 8 = -5x and 3x = 2y -18?
A no intersectionB intersect at (2,-6)
C intersect at (-2,6)D are identical
Solu
tion
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139 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. Set up a system to solve. Which method will you use? (Solving it comes later...)
A graphingB substitutionC elimination So
lutio
n
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140 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many quarters do you have?
Solu
tion
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141 You have 15 coins in your pocket that are either quarters or nickels. They total $2.75. How many nickels do you have?
Solu
tion
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142 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $5.00. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.00. How much does one chocolate chip cookie cost?A $0.50B $0.75C $1.00 D $2.00
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
tion
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143 Mary and Amy had a total of 20 yards of material from which to make costumes. Mary used three times more material to make her costume than Amy used, and 2 yards of material was not used. How many yards of material did Amy use for her costume?
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
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144 The tickets for a dance recital cost $5.00 for adults and $2.00 for children. If the total number of tickets sold was 295 and the total amount collected was $1220, how many adult tickets were sold?
From the New York S ta te Educa tion Department. Office of Assessment Policy, Deve lopment and Adminis tra tion. Inte rne t. Available from www.nysedregents .org/Integra tedAlgebra ; accessed 17, June , 2011.
Solu
tion
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145 A school is selling t-shirts and sweatshirts for a fundraiser. The table shows the number of t-shirts and the number of sweatshirts in each of the three recent orders. The total cost of A and B are given. Each t-shirt has the same cost, and each sweatshirt has the same cost.
The system of equations shown can be used to represent the situation.
Part A: What is the total cost of 1 t-shirt and 1 sweatshirt?
From PARCC sample test
2x + 2y = 38
3x + y = 35{
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146 Part B (continued from previous question)
Select the choices to correctly complete the following statement. In the system of equations, x represents _______ and y represents _______ .
(Type in for x first, then for y.)
A the number of t-shirts in the order
B the number of sweatshirts in the order
C the cost, in dollars, of each t-shirt
D the cost, in dollars, of each sweatshirt
2x + 2y = 38
3x + y = 35{
From PARCC sample test
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147 Part C (continued from previous question)
If the system of equations is graphed in a coordinate plane, what are the coordinates (x , y) of the intersection of two lines?
( ___ , ___)
Students type their answers here
From PARCC sample test
2x + 2y = 38
3x + y = 35{
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148 Part D (continued from previous question)
What is the total cost in dollars, of order C?
$___________
Students type their answers here
From PARCC sample test
2x + 2y = 38
3x + y = 35{
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Instruction
Coordinate Plane
The two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis.
a.k.a.
Coordinate Graphor
Cartesian Plane
Plot lines and points!
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Instruction
EliminationThe process of eliminating one of the
variables in a system of equations.
2x - 3y = -24x + y = 24
System:
Eliminate the y variable
2x - 3y = -24x + y = 24(3)( ) (3)
2x - 3y = -212x + 3y = 72( )+
14x = 70 x = 5
2(5) - 3y = -2 10 - 3y = -2
-3y = -12 y = 4
Solution: (5,4)
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Geometry Theorem
Through any two points in a plane there can be drawn only one line.
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GradeA unit engineers use to measure the
steepness of a hill.
10 feet
25 meters
3 meters
5 feet
25.
5
3
10= 2 grade of
hill is 2.
grade of hill is
The sign warns
cars the hill has a grade of 7.
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Linear Equation
Any equation whose graph is a line.
y = mx + bwhere "b" is the line's y-intercept
and "m" is its slope.
slope intercept
form:point slope
form: y - y1 = m(x - x1)
where "(x1,y1)" is a point on the line
and "m" is its slope.
standard form:
ax + by = cwhere a is non-negative and a and b cannot
both be 0.
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(0,0)
OriginThe point where zero on the x-axis
intersects zero on the y-axis. The point
(0,0).
Used to
graph
coordinates!
(4,-3)
right 4 from origin
down 3 from origin
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ParallelTwo lines that have the same slope and never interesent.
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Perpendicular
Two lines that interset and form a right angle.
RightAngle
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Proportional Relationship
When two quantities have the same relative size.
24
12
if weight is proportional to age, then a weight of 3kg
on the 1st day means it will weigh 6kg on the 2nd day, 9kg on the 3rd day, 30kg on
the 10th day, etc.
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Quadrant Any of the four regions created when the x-axis intersects the y-axis that are usually numbered
with Roman numerals.
II I
III IV
"First Quadrant"
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SlopeHow much a line
rises or falls
y = mx + b
"m" = slope
Steepness of a line
The ratio of a line's rise over its run
"Steepness" and "Position" of a Lineformula for slope: m = y2 - y1
x2 - x1
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Substitution
The process of putting in a value in place of another.
y = 24 - 4x 2x - 3(24 -4x) = -22x - 72 + 12x = -2
14x = 70x = 5
y = 4
2x - 3y = -24x + y = 24 2x - 3y = -2
System: Solution: (5,4)
Check:2(5) - 3(4) = -2
y = 24 - 4(5)
10 - 12 = -2 -2 = -2
4(5) + (4) = 24 20 + 4 = 24
24 = 24
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SystemTwo or more linear equations working together.
y = 2x -3y = x - 1 3x + y = 11
x - 2y = 6
2x + y = 3x - 2y = 4
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System Solution
An ordered pair that will work in each equation.
3x + y = 11x - 2y = 6
system
solution
x = 6 + 2y3(6 + 2y) = 1118 + 6y = 11
6y = -7y = -7 6
x = 6 + 2(-7/6)x = -3 7/10
(-7/6, -3 7/10)
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Table of Values
Numbers or quantities arranged in rows and columns.
rows
columns
x y0
321
1613107
0
321
x y3(x)+7 (x,y)
1613107
3(3)+73(2)+73(1)+73(0)+7
(3,16)(2,13)(1,10)(0,7)
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X-axis Horizontal number line that extends
indefinitely in both directions from zero.
( a , b )
"x-coordinate" (distance away from
origin on x-axis)
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x-intercept
Where a line crosses the x-axis.
None!
(-2,0)
(4,0)
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Y-axisVertical number line that extends
indefinitely in both directions from zero.
( a , b )
"y-coordinate" (distance away from
origin on y-axis)
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y-intercept
Where a line crosses the y- axis.
(0,-6)
(0,4) y = mx + b
"b" = y-intercept (where it crosses the y-axis)
(0,b)
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