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Math 72

3.1 Exploring the Slope of a Line and Rate of Change

What comes to mind when you hear the word slope?

Slope describes the ______________ and the _____________ of a line.

Give an example of where it would be helpful to know the slope of a line?

The slope of a line is the ratio of the _____________ change to the ______________ change.

Slope = run

rise

x

y

changehorizontal

changeveritical

..

..

The slope of a line defines the vertical change relative to the horizontal change of line, as we trace

from left to right along the graph.

Determine the slope of the line.

When finding the slope of a line, select two points on the line that also lie on the intersection of two

grid lines.

Determine the slope of the line.

y y

x x

y y

x x

2

1. Find the slope of each line.

2. How can you decide whether a slope of a line is positive or negative?

A line with a positive slope ________________________________.

A line with a negative slope _______________________________.

3. Study examples 1a, b, and c, the steeper the line, the ___________ the slope.

4. Describe in words how to find the slope of a line?

3

5. Note the grid lines have not been drawn below. Nevertheless, you can still determine the

slope

of each line. Find the slope of each line. Keep track of the method you are using.

6. Determine the slope of the line joining the two points without graphing the lines.

a. (0, 0) (3, 4) b. (1, 2) (3, 5)

c. Describe the method you are using to find the slope.

d. If (x 1 , y 1 ) stands for a point and (x 2 , y 2 ) stands for another point on a line, how

could you find the slope of the line?

The slope of a line passing through two points (x 1 , y 1 ) and (x 2 , y 2 ) is

Slope = run

rise

x

y

changehorizontal

changeveritical

..

..=

12

12

xx

yy

4

7. Find the slope of the line passing through the points (1, 4) and (8, 6).

Slope of a Line Through 1 1,x y and 2 2,x y

Algebraically

2 1

2 1

y ym

x x

for 1 2x x .

ym

x

Verbally

The slope of a line is the

ratio of the change in y

to the change in x.

Numerical Example

1 3

2 1

3 1

4 3

x y

A 1-unit

change in x

produces a 2-

unit change

in y.

For the points

2, 1 and 3,1 ,

1 1

3 2m

2

1m

Graphical Example

Calculate the slope of the line through each pair of points then graph a line that passes through the points.

8. 2,7 and 3,5

9. 1, 8 and 7, 3

y

x

-8

8

-8 8

x

y

-8

8

-8 8

x

y

3,1

1 unit

change in x

2 unit

change in y

2, 1

5

Numerically

m is positive

Verbally

The line slopes _________________________ to the right.

m is negative The line slopes _________________________ to the right.

m is zero The line is _________________________.

m is undefined The line is _________________________.

Using the given point and slope, determine another point on the line and graph the line.

12. Through 0, 3 with m = 1

3

13. Through (0, 1) with 2

3m

-8

8

-8 8

x

y

-8

8

-8 8

x

y

10. 5,3 and 2,3

11. 5,3 and 5, 2

-8

8

-8 8

x

y

-8

8

-8 8

x

y

6

14. Calculate the slope of the line containing the

points in the table.

Rate of change

When we attach units to a slope calculation we say that we have found a rate of change. Rate of change

tells us how one item changes in response to a change in another item.

A few rates that we are familiar with are unemployment rate, death rate, savings rate, crime rate, and

interest rate.

A rate is a ratio that indicates how two quantities change with respect to each other. To calculate a rate, it

is important to keep track of the units being used. Find the slope and include the units.

15. Suppose a student travels at a constant rate on a road trip.

a. Calculate the slope.

b. Find the rate of change.

c. Graph the above data and draw a line that contains the data points. Label the axes.

Recalculate the rate of change by finding that slope of the line graphed and including the units

of measurement.

Graphs allow us to visualize a rate of change. Because graphs make use of 2 axes, they allow us to

visualize how two quantities change with respect to each other. As a rule, the quantity listed in the

numerator appears on the vertical axis and the quantity listed in the denominator appears on the horizontal

axis.Math 72

Time (hours) Distance ( miles)

0 0

1 50

2 100

3 150

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3.2 Discovery Activity

1. Graph: y = 2x – 1

Name the slope:_______

Name the y-intercept:______

2. Graph: y = -2x + 2

Name the slope:_______

Name the y-intercept:______

3. Graph: y = x - 3

Name the slope:_______ Name the y-intercept:______

x y (x, y)

x y (x, y)

x y (x, y)

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Math 72

3.2 Special Forms of Linear Equations in Two Variables

Objective: Use the slope-intercept and point-slope forms of a linear equation.

Slope-Intercept Form

Algebraically

y mx b is the

equation of a line with

slope m and y-intercept

0, b .

Algebraic Example

13

2y x

Verbal Example

This line has slope 1

2

and a y-intercept of

0, 3 .

Graphical Example

1. The line 2

45

y x has a slope of __________ and a

y-intercept of ____________.

2. Graph the line 2

45

y x using the slope and y-intercept.

3. Write the equation in slope-intercept form: y mx b

8 10 20x y

Parallel Lines

Parallel lines have the same slope.

Determine if the lines are parallel.

4. 3

24

f x x and 3

14

f x x

5. 2 3y x and 6 12 24x y

-7

3

-3 7

x

y

0,3

13

2y x

2

1

2,4

9

Objective: Use the special forms of equations for horizontal and vertical lines.

Horizontal and Vertical Lines

Algebraically

Numerical

Example

Graphical Example Verbally

y b is the equation of

a horizontal line with

y-intercept 0, b .

Example: 3y

2 3

1 3

0 3

1 3

2 3

x y

This horizontal line has a y-intercept

of 0, 3 and a slope of 0.

x a is the equation of

a vertical line with

x-intercept , 0a .

Example: 2x

2 2

2 1

2 0

2 1

2 2

x y

This vertical line has an x-intercept of

2, 0 and its slope is undefined.

6. All points on a horizontal line have the same _____-coordinate. This is the reason that the equation of

a horizontal line is of the form _________________________. The slope of a horizontal line is

_____.

7. All points on a vertical line have the same _____-coordinate. This is the reason that the equation of a

vertical line is of the form _________________________. The slope of a vertical line is

_________________________.

Graph each equation by completing a table of values and then give any intercepts. Can you check both of

these on a graphing calculator? If not, why not?

8. Equation: 3x

Graph:

x y

x-intercept: ______

y-intercept: ______

Slope: _______

9. Equation: 2y

Graph:

x y

x-intercept: ______

y-intercept: ______

Slope: _______

-4

4

-4 4

y

x

-4

4

-4 4

y

x

-5

5

-5 5x

y

-5

5

-5 5x

y

3y

2x

10

10. Write the equation of the line passing through the point (1, 2) with a slope of 3.

11. Write the equation of the line passing through the point (2, 5) with a slope of -1.

Write the equation of the line passing through the given point with specified slope. Write the answer in

slope-intercept form.

12. 4, 2 ,

3

4m

13. 4, 3 , 2

3m

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14. Write the equation of a line passing through 2, 4 and 1, 3 .

15. Write the equation of a line passing through 4, 4 and 1, 2 .

General Form

The general form, Ax By C , of an equation is useful for writing linear equations without fractions.

Write each equation in general form.

16. 17.

13 5

x y

1 4

3 3y x

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Math 72

3.3 Solving Systems of Linear Equations in Two Variables Graphically and Numerically

Objective: Determine the point where two lines intersect.

When we refer to two or more equations at the same time we refer to this as a system of

equations. A point where two lines intersect is called a ____________ of a system of linear

equations. This point of intersection is an ordered pair that makes both equations ____________

at the same time.

1. Check each point to test if it is a solution of each equation:

Point Solution of 7y x ? (yes/no) Solution of 3 5y x ? (yes/no)

6,1

1, 2

3,4

(b) Which point would you conclude is a solution of both equations?

(c) Enter 1 7Y x and

2 3 5Y x using a window of 10,10,1 by 10,10,1 on your

calculator and press 2nd

, TRACE, 5, ENTER, ENTER, ENTER. Does this support

your conclusion?

Objective: Solve a system of linear equations using graphs and tables.

2. Determine whether the ordered pair 2, 2 is a solution of the system of linear equations

3 2

4

y x

y x

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3. Graph the system:2

4

3

xy

y x

(a) Determine the point of intersection.

(b) Verify that this point satisfies both equations.

(c) Use the Intersection feature on your calculator to confirm the above results.

4. Solve each equation for y. Graph each equation and determine the solution of the system of linear

equations.

Then use your calculator to check your work.

5

3 3

x y

x y

Solution: __________________

-5

5

-5 5

x

y

-5

5

-5 5

x

y

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Objective: Identify inconsistent systems and systems of dependent linear equations

A solution of a system of equations is an _____________________ _____________________, ,x y ,

that satisfies each equation in the system.

Solutions Sets for a System of Two Linear Equations:

In graphing a system of two linear equations, we have three possible outcomes:

One solution: The equations describe lines intersecting at exactly one point. This system of equations

has exactly ______ ____________--the ordered pair at the ________ ___ __________.

Consistent system & Independent equations

No solution: The equations describe parallel lines. The graphs have ____ _________ of intersection;

hence, the system has ____ real number ___________.

Inconsistent system & Independent equations

Infinite number of solutions: The equations describe the same line. The system has an _____________

number of _______________ because the coordinates of every point on the graph of the first equation

_____________ the second equation.

Consistent system & Dependent equations

5. Classify each system of equations.

A. A consistent system of independent linear equations having exactly one solution.

B. An inconsistent system of linear equations having no solution.

C. A consistent system of dependent linear equations having an infinite number of solutions.

(a) 2

3

y x

y x

________ (b)

2

2

y x

y x

________ (c)

2 1

1

y x

y x

________

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Math 72

3.4 Solving Systems of Linear Equations in Two Variables by the Substitution Method

Objective: Solve a system of linear equations by the substitution method.

Substitution Method

Step 1. Solve one of the equations for one

______________________ in terms of

the other ______________________

Example: 6 5 12

2 4

x y

x y

Step 2. ______________________ the

expression obtained in Step 1 into the

other equation (eliminating one of the

variables), and solve the resulting

equation.

Step 3. Substitute the ______________________

obtained in Step 2 into the equation

obtained in Step 1 (back-substitution) to

find the value of the other variable.

The ordered pair obtained in Steps 2 and 3 is the

solution that should check in both equations.

Solve each system using the substitution method.

1. 1

2 5

x y

x y

2. 3 5

2 4

x y

x y

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Solve each system using the substitution method.

3. 3 2 6

4 8

x y

x y

4.

13 4

12 6

3

x y

y x

5. 2 3

6 3 3

y x

x y

6. 6 2 8

3 4

x y

y x

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7. The costs for renting a rug-shampooing machine from two different rental companies are given by the

graphs shown below. The graph of 1 1( )y f x gives the cost by Dependable Rental Company based

upon the number of hours of use. The graph of 2 2 ( )y f x gives the cost by Anytime Rental

Company based upon the number of hours of use.

(a) Use the y-intercept and an additional point to

determine the equation of the line for the

Dependable Rental Company.

(b) Use the y-intercept and an additional point to determine the equation of the line for the Anytime

Rental Company.

(c) Solve the system of equations using the substitution method.

(d) Interpret the meaning of the x- and y-coordinates of this solution.

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6

y

x

$ Cost

Hours

1 1y f x

2 2y f x

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3.5 Lecture Guide:

Solving Systems of Linear Equations in Two Variables by the Addition Method

Application: The cost of renting a car from Rent-a-Wreck for a day is a $50 base fee plus 10 cents per

mile. Lemon Renting charges a $30 initial fee and 15 cents per mile. For how many miles will the cost

be the same? What is that cost?

Let’s set up the problem:

How do we solve this?......Let’s learn a new method and come back to this problem after we

practice the new method.

Objective: Solve a system of linear equations by the addition method.

Addition Method

Step 1. Write both equations in the

__________________ form Ax By C

.

Example: x – y = 3 x + y = 5

Step 2. If necessary, multiply each equation by a

constant so that the equations have one

variable for which the coefficients are

additive __________________.

Step 3. Add the new equations to

__________________ a variable, and

then solve the resulting equivalent

equation.

Step 4. __________________ this value into one

of the original equations, and solve for

the other variable.

The ordered pair obtained in Steps 3 and 4 is the

solution that should check in both equations.

Solve each system using the addition method.

1. 2x + y = 10 2. 2x – 7y = 1 3x – y = 5 x + 5y = 9

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Solve each system using the addition method.

3. 6x + 5y = 12

2x + y = 4 4. 2x – 3y = 14

3x = 5y + 22

5. 4x – 8y = 20

6. 5x – 6y = 4

3x – 6y = 15 10x = 12y + 5

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7. Now, let’s go back to our application: The cost of renting a car from Rent-a-Wreck is a $50

base fee plus 10 cents per mile. Lemon Renting charges a $30 initial fee and 15 cents per mile.

For how many miles will the cost be the same? What is that cost?

(a) Define the variables.

b) Write a system of algebraic equations.

c) Solve the system algebraically using the Addition Method.

d) Answer in a sentence.

8. Find two numbers whose sum is 60 and whose difference is 14.

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3.6 Lecture Guide: More Applications of Linear Systems

Objective: Use systems of linear equations to solve word problems.

Strategy for Solving Word Problems

Step 1. Read the problem carefully to determine what you are being asked to find. Step 2. Select a variable to represent each unknown quantity. Specify precisely what each variable

represents and note any restrictions on each variable. Step 3. If necessary, make a sketch and translate the problem into word equations. Then translate the

word equations into a system of algebraic equations. Step 4. Solve the equation or the system of equations, and answer the question asked by the problem.

Step 5. Check the reasonableness of your answer.

1. Marlon is making a rectangular picture frame out of wood molding. He has 42 inches of molding to create

the frame. If the width of the frame must be 5 inches less than the length, what will be the dimensions of

the frame? Will a 9 inch by 12 inch picture fit in this frame?

(a) Select a variable to represent each of the unknown quantities and identify each variable, including the units

of measurement.

(b) Write a system of algebraic equations.

(c) Solve this system of equations.

(d) Write a sentence that answers the problem.

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Rate Principle

Amount = Rate×Base A R B

Applications of the Rate Principle

1. Variable cost = Cost peritem × Numberof items

2. Interest = Principal invested×Rate×Time

Mixture Principle for Two Ingredients

Amount in first + Amount in second = Amount in mixture

Applications of the Mixture Principle 1. Variable cost + Fixed cost = Total cost 2. Interest on one investment + Interest on second investment = Total interest

2. Ashley invested money in two different accounts. One investment was at 10% simple interest and the other

was at 12% simple interest. The amount invested at 10% was $1500.00 more than the amount invested at

12%. The total interest earned was $480.00. How much money did Ashley invest in each account?

3. A student saving for college was given $6,000 by her grandparents. She invested part of this

money in a savings account that earned interest at the rate of 3% per year. The rest was invested

in a bond that paid interest at the rate of 5.5% per year. If the combined interest at the end of one

year was $305, how much was invested at each rate?

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4. You are offered two different sales jobs. The first company offers a straight commission of 3%

of the sales. The second company offers a salary of $430 per week plus 2% of the sales. How

much would you have to sell in a week in order for the straight commission offer to be at least

as good?

5. If 104 people attend a concert and tickets for adults cost $3.75 while tickets for children cost

$3.50 and total receipts for the concert was $377, how many of each went to the concert?

6. Two parts that brace a portion of a playground slide are joined at a common point. The two

angles that form at this common vertex must be complementary (sum is 90 degrees). If

the larger angle is 20° more than the smaller angle, determine the number of degrees in each

angle.