Graphing Linear Equations Section 1.2. Lehmann, Intermediate Algebra, 3ed Section 1.2 Consider the...
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Transcript of Graphing Linear Equations Section 1.2. Lehmann, Intermediate Algebra, 3ed Section 1.2 Consider the...
Lehmann, Intermediate Algebra, 3edSection 1.2
Consider the equation . Let’s find y when
So, when , which cab be represented by the ordered pair
Slide 2
Definition of Solution, Satisfy, and Solution Set
2 5y x 3.x
2 5
2 3 5
6 5
1
y x
y
Original Equation.
Substitute 3 for x.
Multiply before subtracting.
Subtract.1y 3x
3,1 .
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
For an ordered pair , we write the value of the independent variable in the first (left) position and the value of the dependent variable in the second (right) position.
•The numbers a and b are called coordinates.
•For the x-coordinate is 3 and the y-coordinate is 1.
Slide 3
Definition of Solution, Satisfy, and Solution Set
,a b
3,1 ,
Definition
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
The equation becomes a true statement when we substitute 3 for x-coordinate and 1 for y-coordinate.
Slide 4
Definition of Solution, Satisfy, and Solution Set
2 5y x
?
?
2 5
1 2 3 5
1 1
true
y x
Original Equation.
Substitute 3 for x and 1 for y.
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
• An ordered pair is a solution of an equation in terms of x and y if the equation becomes a true statement when a is substituted for x and b is substituted for y.
• We say satisfies the equation.
• The solution set of the equation is the set of all solution of the equation.
Slide 5
Definition of Solution, Satisfy, and Solution Set
,a b
,a b
Definition
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
Find five solutions to the equation and plot them in the coordinate system (on the right).
Slide 6
Graphing an Equation
2 1,y x
Example
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
We begin be arbitrarily choosing the values 0, 1, and 2 to substitute for x:
The ordered pairs and are also solutions.
Slide 7
Graphing an Equation
2 0 1 2 1 1 2 2 1
0 1 2 1 4 1
1 1 3
Solution: 0,1 Solution: 1, 1 Solution: 2, 3
y y y
2,5 1,3
Solution
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
• Create a table of solutions
x y
-2 5-1 3 0 1 1 -1
2 -3
Slide 8
Graphing an Equation
• Plot the solutions
• Points form a linear line.
Solution Continued
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
• Every point on the line is a solution to the equation
Slide 9
Graphing an Equation
2 1y x
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
• The point lies on the line • Should satisfy the
equations• Whereas is not
on the line• Thus should not
satisfy the equation
Slide 10
Graphing an Equation
3, 5
2,4
2 1y x
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2 Slide 11
Graphing an Equation
?
?
2 1
4 2 2 1
4 3
false
y x
Original Equation.
Substitute 2 for x and 4 for y.
• The is not a solution to the equation 2,4
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
Use ZDecimal on a graphing calculator.
•To enter press (–) 2 X,T,ϴ,n + 1. The key – is used for subtraction, and the key . (–) is used for negative numbers as well as taking the opposite.
Slide 12
Graphing an Equation
2 1,y x
Calculator
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3edSection 1.2
The graph of an equation in two variables is the set of points that correspond to all solutions of the equation.
In the last example we found that the equation . is a line. Notice that the equation . is of the form (where and ).
Slide 13
Definition: Graph
2 1,y x 2 1,y x
y mx b 2m 1b
Definition
Definition of Solution, Satisfy, and Solution Set
Lehmann, Intermediate Algebra, 3ed
Equations of the form
Section 1.2
If an equation can be put into the form
where m and b are constants, then the graph of the equation is a line.
Slide 14
Graphs of Linear Equations
y mx b
y mx b
What is m and b for the equations3
2, 3 and 3?2
y x y x y
Example
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3ed
is of the form : and
is of the form because we write the equation as (so and ).
is of the form because we write the equation as (so and ).
Sketch the graph of the equation
Graphing Linear Equations
32
2y x y mx b 2b
32
m
2y x y mx b 2 0y x 0b 2m
3b 3y
0 3y x 0m y mx b
30 6 5 0.x y
Definition
Example
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
First we solve for y
Slide 16
Graphing Linear Equations
30 6 12 0
30 6 12
6 30 12
6 30 126 6 6
5 2
x y
x y
y x
y x
y x
Original Equation.
Subtract 12 from both sides.
Subtract 30x from both sides.
Simplify.
Divide both sides by 6.
Solution
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
• . is of the form• The graph of the equation is a line• Find 2 points of the line• Plot the two points• Sketch the line• Find a third point•Verify that the third point lies on the line
Slide 17
Graphing Linear Equations
5 2y x y mx b
Graphs of Linear Equations
Solution Continued
Lehmann, Intermediate Algebra, 3edSection 1.2 Slide 18
Graphing Linear Equations
Table of solutions
x y
2 0 3 30
1 2 1 3 1
2 2 2 3 1
Solution Continued
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
• Enter for y1.
• Use Zstandard followed by Zsquare.
• The graph is correct assuming that y was isolated correctly.
Slide 19
Graphing Linear Equations
5 2x Graphing Calculator
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
Sketch the graph of
•Use the distributive property on the left-hand side.•Collect like terms.•Isolate y.
Slide 20
Using the Distributive Law to Help Graph a Linear Equation
3 2 5 2 3 8 .y x x Example
Solution
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2 Slide 21
Using the Distributive Law to Help Graph a Linear Equation
3 2 5 2 3 8
6 15 6 3
6 15 15 6 3 15
6 6 126 6 6
2
y x x
y x
y x
y x
y x
Original equation
Distributive property
Add 15 to both sides.
Divide both sides by 6.
Simplify.
Solution Continued
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2 Slide 22
Using the Distributive Law to Help Graph a Linear Equation
Table of solutions
x y
0 2 20
1 1 2 1
2 2 2 0
Solution Continued
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
Sketch a graph of
•Avoid fraction values for y
•Use even values for x
Slide 23
Graphing an Equation That Contains Fractions
11.
2y x
Example
Solution
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3ed
Table of solutions
x y
0
2
4
Section 1.2 Slide 24
Graphing an Equation That Contains Fractions
10 1 1
21
2 1 021
4 1 12
Solution Continued
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
Use Zdecimal to verify the solution.
Slide 25
Graphing an Equation That Contains Fractions
Graphing Calculator
Graphs of Linear Equations
Lehmann, Intermediate Algebra, 3edSection 1.2
Sometimes we find intercepts to graph a line.
• x-intercept is on the y-axis, so y = 0
• y-intercepts in on the x-axis, so x = 0
Slide 26
Property
• For an equation containing the variables x and y
• x-intercept: Substitute y = 0 and solve for x
• y-intercept: Substitute x = 0 and solve for y
Directions
Finding Intercepts of a Graph
Lehmann, Intermediate Algebra, 3edSection 1.2
Use intercepts to sketch a graph of
Slide 27
Using Intercepts to Sketch a Graph
2 4.y x
x-intercept: Set y = 0.
0 2 4
4 2
4 22 22
x
x
x
x
Simplify.
Substitute 0 for y.
Subtract both sides by 4.
Divide both sides by -2.
Example
Solution
Finding Intercepts of a Graph
Lehmann, Intermediate Algebra, 3edSection 1.2
y-intercept: Set x = 0.
So, the x-intercept is and y-intercept is
Slide 28
Using Intercepts to Sketch a Graph
2 0 4
4
y
y
Set x=0.
Simplify.
0,4 . 2,0
Solution Continued
Finding Intercepts of a Graph
Lehmann, Intermediate Algebra, 3edSection 1.2
Use ZStandard followed by ZSquare.
Use “zero” to verify the x-intercept.
Slide 29
Using Intercepts to Sketch a Graph
Use TRACE to verify the y-intercept.
Graphing Calculator
Finding Intercepts of a Graph
Lehmann, Intermediate Algebra, 3edSection 1.2
Graph the equation of
Slide 30
Graphing a Vertical Line
3.x
x y
3 53 33 13 -13 -3
Notice that the values of x must be 3, but y can have any value. Some solutions are listed to the left.
Example
Solution
Finding Intercepts of a Graph
Lehmann, Intermediate Algebra, 3edSection 1.2
Graph the equation of
Slide 31
Graphing a Horizontal Line
5.y
x y
–2 –5 –1 –5 0 –5 1 –5 2 –5
Notice that the values of y must be –5, but x can have any value. Some solutions are listed to the left.
Example
Solution
Vertical and Horizontal Lines
Lehmann, Intermediate Algebra, 3edSection 1.2
Use ZStandard to verify the graph.
Slide 32
Graphing a Horizontal Line
Graphing Calculator
Vertical and Horizontal Lines
Lehmann, Intermediate Algebra, 3edSection 1.2
If a and b are constants:
• An equation that can be put into the form . has a vertical line as its graph
• An equation that can be put into the form .has a horizontal line as its graph
Slide 33
Vertical and Horizontal Line Property
y b
x a
Property
Vertical and Horizontal Lines
Lehmann, Intermediate Algebra, 3edSection 1.2
In an equation can be put into either form
where m, a, and b are constants, then the graph of the equation is a line. We call such an equation a linear equation in two variables.
Slide 34
Vertical and Horizontal Line Property
ory mx b x a
Property
Vertical and Horizontal Lines