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Constant Rate of Change

Lesson 1

Marcus can download two songs from the Internet each minute. This is shown by the table below.

Real World Link

a. Compare the change in number of songs y to the change in time x. What is the rate of change?The number of songs increases by 2, and the time

increases by 1.

b. Graph the ordered pairs. Describe the pattern.The points appear to make a straight line.

1. Relationships that have a straight-line graph

2. The rate of change is the same – “constant rate of change”

Linear Relationships

The balance in an account after several transactions is shown. Is the relationship linear? If so, find the constant rate of change.

Since the rate is constant, this shows a linear relationship. The constant rate of change is or -10. Each transaction involved a $10 withdrawal.

Example 1

a. b.

Got it? 1

No, this is not a linear relationship. It does not have a constant

rate of change.

Yes; the points make a line so this is a linear

relationship. The constant rate of

change is -2.5 minutes per volunteer.

Words: Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change.

Symbols: is constant and is also constant.

Example: there is a constant rate of change and

Proportional Linear Relationships

21=42=63=84

Use the table to determine if there is a proportional linear relationship between the temperature in degrees and Fahrenheit and a temperature in degrees Celsius. Explain your reasoning.

Example 2

Since the rate of change is constant, this is a linear relationship.

Since the ratios are not equal, degrees Fahrenheit

and Celsius are not proportional.

Use the table to determine if there is proportional linear relationship between the mass of an object in kilograms and the weight of the object in pounds. Explain your reasoning.

Got it? 2

Weight (Ibs)

Mass (kg)

20 9

40 18

60 27

80 36

It is a proportional linear relationship.

The ratio is a constant , and the rate of change is .

{Slope

Lesson 2

Slope =

Find the slope of the treadmill.

Example 1

Slope =

=

=

The slope of the treadmill is .

A hiking trail rises 6 feet for every horizontal change of 100 feet. What is the slope of the hiking trail?

Slope =

=

The slope of the trail is .

Got it? 1

The graph shows the cost of muffins at a bake sale. Find the slope of the line.

Example 2

The vertical change is 2 and the horizontal change is 1.

The slope is or 2.

The table shows the number of pages Garrett has left to read after a certain number of minutes. The points lie on a line. Find the slope of the line.

Example 3

Slope =

Choose any points: (1, 12) and (3, 9)

Slope =

The slope is - .

a. b.

Got it? 2 & 3

The slope is .

The slope is .

Symbols:m = slope

m =

Slope Formula

Find the slope of the line that passes through R(1, 2), S(-4, 3).

Example 4

m =

m =

m =

The slope of the line is - .

a. A(2, 2), B(5, 3) b. J(-7, -4), K(-3, -2)

Got it? 4

1335

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Equations in y = mx Form

Lesson 3

Find the slope of this graph.

m =

m =

Solve for y.

y = mx

Words: “y varies directly with x”

Symbols: m = or y = mx, where m is the constant or slope.

Example: y = 3x

Direct Variation

Graph:

The amount of money Robin earns while babysitting varies directly with the time as shown in the graph. Determine the amount that Robin earns per hour.

Example 1

Find “nice” points on the graph to find the constant of variation.

(2, 15) and (4, 30)

m = m = = 7.5 m = =

7.5

So Robin earns $7.5 and hour babysitting.

Two minutes after a skydiver opens his parachute, he has descended 1,900 feet. After 5 minutes, he descends 4,750 feet. If the distance varies directly with the time, at what rate is the skydiver descending?

Hint: Find the two points on the line.

-950 feet/minute

Got it? 1

A cyclist can ride 3 miles in 0.25 hour. Assume that the distance biked in miles varies directly with time in hours x. This situation can be represented by y = 12x. Graph the equation. How far can the cyclist ride per hour?

Make a table of values.

Example 2

Graph the values. The slope is 12 since the equation is y = 12x.

12 miles per hour

A grocery store sells 6 oranges for $2. Assume that the cost of the oranges varies directly with the number of oranges. This situation can be represented by y = x. Graph the equation. What is the cost per orange?

about $0.33 per orange

Got it? 2

In a proportional relationship, how is the unit rate represented on a graph?

It is the slope.

When comparing two different direct variation equations, what’s the difference?

y = 3x y = 12x

The rate or slope.

Comparing Direct Variation

The distance d in miles covered by a rabbit in t hours can be represented by d = 35t. The distance covered by a grizzly bear is shown on the graph. Which animal is faster, or has the fastest rate?

Example 3

Rabbit:

d = 35t, so the rate is 35.

Grizzly Bear:

Find the slope of the line.

The rate is 30.35 is greater than 30, so

the rabbit is faster.

Damon’s earnings for four weeks from a part time job are shown in the table. Assume that his earnings vary directly with the number of hours worked.

He can take a job that will pay him $7.35 per hour. What job is the better pay? Explain.

His current job is a better pay. He currently earns $7.50.

Got it? 3

A 3-year old dog is often considered to be 21 in human years. Assume that the equivalent age in human years y varies directly with its age as a dog x. Write and solve a direct variation equation to find the human-year age of a dog that is 6 years old.

Example 4

We know that when y is 21, x is 3.

y = mx21 = m(3)

m = 7

The rate is 7.

y = 7x

y = 7(6)y = 42

A dog that is 6 years old has an equivalent human age of

42.

a. A charter bus travels 210 miles in 3.5 hours. Assume the distance traveled is directly proportional to the time traveled. Write and solve a direct variation equation to find how far the bus will travel in 6 hours.

y = 60x 360 miles

b. A Monarch butterfly can fly 93 miles in 15 hours. Assume the distance traveled is directly proportional to the time traveled. Write and solve a direct variation equation to find how far the Monarch butterfly will travel in 24 hours.

y = 9.3x 148.8 miles

Got it? 4

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Slope-Intercept Form

Lesson 4

Slope-Intercept Form

State the slope and the y-intercept of the graph of the equation y = x – 4.

y = mx + b, where m is slope and b is y-intercepty = x – 4

m = and b = -4The slope is and the y-intercept is -4.

Example 1

Find the slope and y-intercept for each equation.

a. y = -5x + 3-5; 3

b. y = x – 6

c. y = -x + 5-1; 5

Got it? 1

Write an equation of a line in slope-intercept form if you know that the slope is -3 and the y-intercept is -4.

y = mx + by = -3x + (-4) or y = -3x – 4

Write an equation of a line in slope-intercept form from the graph below.

Examples 2 & 3

y = mx + bthe slope is 2 and the

y-intercept is 4y = -x + 4

a. Write an equation in slope-intercept form for the graph.

Got it? 2 & 3

b. Write an equation in slope-intercept form with a slope of and a y-intercept of -3.

y = x - 2

y = x - 3

Student Council is selling T-shirts during spirit week. It costs $20 for the design and $5 to print each shirt. The cost y to print x shirts is given by y = 5x + 20. Graph this equation using the slope and y-intercept.

Step 1: Find the slope and y-intercept. slope = 5 y-intercept = 20

Step 2: Graph the y-intercept (0, 20)

Step 3: Go up 5 and over 1 to find another point.

Example 4

Student Council is selling T-shirts during spirit week. It costs $20 for the design and $5 to print each shirt. The cost y to print x shirts is given by y = 5x + 20. Interpret the slope and y-intercept.

The slope represents the cost of each T-shirt. The y-intercept is the one time fee of $20 for the

design.

Example 5

A taxi fare y can be determined by the equation y = 0.50x + 3.5, where x is the number of miles traveled.

a. Graph this equation.

b. Interpret the slope and y-intercept. The slope is the $0.50 per mile and the y-intercept is the

initial charge of $3.50.

Got it? 4 & 5

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Graph a Line Using Intercepts

Lesson 5

x and y intercepts

State the x- and y-intercepts of y = 1.5x – 9. Then use the intercepts to graph the equation.

Example 1

STEP 1: Find the y-intercept.y-intercept is -9.

STEP 2: Find the x-intercept, let y = 0.

0 = 1.5x – 99 = 1.5x

6 = x

STEP 3: Graph the two intercepts and connect to make a line.

Graph these equations by using the x- and y- intercepts.a. y = x + 5 b. y = x + 3

Got it? 1

Standard form is when the equation is Ax + By = C, and A, B, and C are integers. A, B, and C can NOT be fractions or decimals.

Take the equation: 60x + 15y = 4,740

A = 60 B = 15 C = 4,740

Standard Form

Mauldlin Middle School wants to make $4,740 from yearbooks. Print yearbooks x cost $60 and digital yearbooks y cost $15. This can be represented by the equation 60x + 15y = 4,740.

Use the x- and y-intercepts to graph the equation. To find the x-intercept, let y = 0. To find the y-intercept, let x = 0

Example 2

60x + 15y = 474060x + 15(0) = 4740

60x = 4740x = 79

60x + 15y = 474060(0) + 15y = 4740

15y = 4740y = 316

Graph the equation, using the intercepts, and interpret the x- and y- intercepts.

Example 3

The x-intercept is at the point (79,0). This means

they can sell 79 print yearbooks and still earn

$4,740.The y-intercept is at the

point (0, 316). This means they can sell 316 digital yearbooks and still earn

$4,740.

Mr. Davis spent $230 on lunch for his class. Sandwiches x cost $6 and drinks y cost $2. This can be represented by the equation 6x + 2y = 230. Graph and interpret the x- and y- intercepts.

Got it? 2 & 3

The x-intercept of 38.33 means that if Mr. Davis

bought 38.33 sandwiches, then the total would be $230.

The y-intercept of 115 means that if he bought

115 drinks the total would be $230.

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Write Linear Functions

Lesson 6

Point-Slope Form

Write an equation in point-slope form for the line that passes through (-2, 3) with a slope of 4.

y – y1 = m(x – x1)

y – 3 = 4(x – (-2))y – 3 = 4(x + 2)

Example 1

Write an equation in slope-intercept form for the line that passes through (-2, 3) with a slope of 4.

y = mx + b3 = 4(-2) + b

3 = -8 + b11 = b

y = 4x + 11

Example 2

Write an equation in point-slope form and slope-intercept form for a line that has a slope of - and a point (-1, 2).

Got it? 1 & 2

Point-Slope Form:y – y1 = m(x – x1)

y – 2 = (x +1)

Slope-Intercept Form:y = mx + b

y = x +

STEP 1: Find the slope.

STEP 2: Use one of the points and the slope to make an equation in point-slope form.

STEP 3: Use one of the points and slope to make an equation in slope-intercept form.

STEP 4: Find the y-intercept.

Writing a Linear Equation from Two Points

Write an equation in point-slope form and slope-intercept form for a line that passes through (8, 1) and (-2, 9).

Example 3

Point-Slope Form:Slope =

Use the point (8, 1)

y – 1 = (x – 8)

Slope-Intercept Form:y = mx + b

y = x + b

Use the point (8, 1)

1 = (8) + b

b =

Write an equation in point-slope form and slope-intercept form for a line that passes through (3, 0) and (6, -3).

Got it? 3

Point-Slope Form:

y + 3 = (x – 6)

ory + 0 = -1(x – 3)

Slope-Intercept Form:

y = -1x + 3

or

y = -x + 3

The cost of assistance dog training sessions is shown in the table. Write an equation in point-slope form to represent the cost y of attending x dog training sessions.

Find the slope. m =

Use the point (5, 165).y – 165 = 25(x – 5)

Example 4

The cost for making spirit buttons is shown in the table. Write an equation in point-slope form to represent the cost of y of making x buttons.

Find the slope. m =

Use the point (100, 25).y – 25 = (x – 100)

Got it? 4

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Solve Systems of Equation by Graphing

Lesson 7

System of Equations:Two or more equations with the same set of

variables.

The solution to a system is when they lines cross at a certain point.

System of Equations

The solution for this system of equations is

(3, 8).

Solve the system y = -2x – 3 and y = 2x + 5 by graphing.

Graph each line on the same coordinate plane.

Example 1

The solution is (-2, 1)

y = -2x – 31 = -2(-2) – 3

1 = 1

y = 2x + 51 = 2(-2) + 5

1 = 1

Graph to find the solution.

y = x – 1 y = 2x – 2

Got it? 1

Gregory’s Motorsports has motorcycles (two wheels) and ATV’s (four wheels) in stock. The store has a total of 45 vehicles, that together, have 130 wheels.

Write a system of equations that represent this situation.

Let x be the # of motorcycles and y be the # of ATV’s. x + y = 45

and2x + 4y = 130

Example 2

Gregory’s Motorsports has motorcycles (two wheels) and ATV’s (four wheels) in stock. The store has a total of 45 vehicles, that together, have 130 wheels.

Solve the system of equations. Interpret the solution. x + y = 45

2x + 4y = 130

Graph the equations.

Example 3

The store has 20 motorcycles and 25

ATV’s.

Creative Crafts gives scrapbooking lesson for $15 per hour plus a $10 supply charge. Scrapbooks Incorporated gives lessons for $20 per hour with no additional charges. Write and solve a system of equations that represents the situation. Interpret the situation.

15x + 10 = y20x = y

The solution is (2, 40). If you take lessons for 2 hoursthe cost at both stores are the same.

Got it? 2 & 3

If the lines intersect, there is one solution.

If the lines are parallel, there are no solutions.

If the lines are the same, there are infinitely many solutions.

Number of Solutions

Solve the system by graphing. y = 2x + 1y = 2x – 3

Since the lines are parallel, there is no solution.

Example 4

Solve the system by graphing. y = 2x + 1

y - 3 = 2x – 2

Since the lines are the same, there are infinitely many solutions.

Example 5

Solve each system of equations by graphing.

a. y = x + 3 b. y – x = 1 3y = 2x + 12 y = x – 2 + 3

Got it? 4 & 5

A system of equations consists of two lines. One line passes through (2, 3) and (0, 5). The other line passes through (1, 1) and (0, -1). Determine if the system has one solution, no solution, or infinite number of solutions.

Carefully graph the points and make the two lines. The lines appear to cross at (2, 3).

Example 6

(2, 3) and (0, 5)Slope = -1Equation:

y = -1x + 5

3 = -2 + 53 = 3

(1, 1) and (0, -1)

Slope = 2Equation:y = 2x – 1

3 = 2(2) – 13 = 3

A system of equations consists of two lines. One line passes through (0, 2) and (1, 4). The other line passes through (0, -1) and (1, 1). Determine if the system has one solution, no solution, or infinite number of solutions.

No solution

Got it? 6

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Solve Systems of Equations Algebraically

Lesson 8

Solve the system of equations algebraically.y = x – 3y = 2x

Since y = 2x, then you can substitute 2x in for the first equation.

y = x – 32x = x – 3

x = -3

When x = -3, then y is -6. The solution is (-3, -6)

Example 1

Solve each system of equations algebraically.

a. y = x + 4 b. y = x - 6 y = 2 y = 3x

Got it? 1

(-2, 2) (-3, -9)

Solve the system of equations algebraically.y = 3x + 8

8x + 4y = 12

Example 2

8x + 4(3x + 8) = 12Use the Distributive

Property8x + 12x + 32 = 12

20x + 32 =1220x = -20

x = -1

x = -1

y = 3(-1) + 8y = -3 + 8

y = 5

The solution is (-1, 5).

Solve each system of equations algebraically.

a. y = 2x + 1 b. 2x + 5y = 44 3x + 4y = 26 y = 6x – 4

Got it? 2

(2, 5) (2, 8)

A total of 75 cookies and cakes were donated for a bake sale to raise money for the football team. There were four times as many cookies donated as cakes.

Write a system of equations to represent this situation.

y = 4xx + y = 75

Example 3

Solve the system of equations from Example 3 algebraically.

x + y = 75y = 4x

x + 4x = 755x = 75x = 15

When x is 15, y is 60. The solution is (15, 60). This means that 15 cakes and 60 cookies were donated to the bake

sale.

Example 4

Mr. Thomas cooked 45 hamburgers and hot dogs at a cookout. He cooked twice as much hot dogs than hamburgers.

a. Write a system of equations that represents this situation.

y = 2xx + y = 45

b. Solve the system algebraically and interpret the solution. (15, 30)

This means that he cooked 15 hamburgers and 30 hot dogs.

Got it? 3 & 4