Section 2.3

Graphs of Linear Equations in Two Variables

Objective: Check possible solutions of a linear equation.

2.3 Lecture Guide: Graphs of Linear Equations in Two Variables

1. Before we check the solution of a linear equation in two variables, we will review how we check a solution to a linear equation in one variable.

(a)

(b)

Is a solution of 3 9 5x x ? 6x

Is a solution of 3 9 5x x ? 7x

We will now check a solution of a linear equation in two variables.

Solution of a linear equation y mx b :

A solution of a linear equation of the form y mx b is an ordered pair ,x y that makes the equation a____________ statement.

(a)

(b)

(c)

2. Test each ordered pair to determine whether it is a solution of the equation 2 1y x

1, 3

2,1

0, 1

2 1y x 3. Graph on your calculator using the ZOOM 4window. Use the TRACE feature to enter the x-coordinates to check the points from question 2.

4. Any ordered pair that is a ____________ of a linear equation will lie on the ____________.

5. Consider the graph below. (a) Give the coordinates of the points A, B, and C.

,A

,B

,C

-10-9-8-7-6-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

xA

B

C

5. Consider the graph below.

(b) Test each of these points

in the equation 4

43

y x

to determine whether it is a solution of this equation.

-10-9-8-7-6-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

xA

B

C

5. Consider the graph below.

(c) Would you say that this graph is or is not the graph of

44

3y x ?

-10-9-8-7-6-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

xA

B

C

(Circle the correct choice.) Explain.

Objective: Determine the intercepts from a graph.

x- and y- intercepts:

The x-intercept of a graph is the point with a y-coordinate of ______.

The y-intercept of a graph is the point with an x-coordinate of ______.

Determine the intercepts of each graph.

6.

x-intercept:_________

y-intercept:_________

-3

-2

-1

0

1

2

3

4

5

6

7

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

7. Recall that the x-intercept of a graph is the point where the graph crosses the ___-axis. Earlier, we stated that all points on this axis have a y-coordinate of ______. The y-intercept is the point where the graph crosses the ___-axis. Earlier, we stated that all points on this axis have an x-coordinate of ______.

8. In the graph below the input x is the number of units produced by a machine in a factory. The output y is the profit made by the sale of these units when they are produced. Determine the intercepts and interpret the meaning of each.

-$300

-$200

-$100

$0

$100

$200

$300

$400

$500

-10 0 10 20 30 40 50 60 70 x (units)

y (Profit)(a) x-intercept: ____________

(b) Meaning of x-intercept:

(c) y-intercept: ____________

(d) Meaning of y-intercept:

9. Use a graphing calculator to graph 3 6y x 1,5,1 7,1,1

using a by

x -intercept: __________

y -intercept: __________

viewing window of .(See Calculator Perspective 2.3.3 for help changing the viewing window.)

(a) Draw a rough sketch of your calculator graph and estimate the x- and y-intercepts.

-7

-6

-5

-4

-3

-2

-1

0

1

-1 0 1 2 3 4 5

y

x

9. Use a graphing calculator to graph 3 6y x 1,5,1 7,1,1

using a by viewing window of .

(See Calculator Perspective 2.3.3 for help changing the viewing window.)

(b) Create a table using TblStart = 1 , Tbl 1 . Does this table confirm your estimates?

–1

0

1

2

3

4

5

x1Y

10. A graphing calculator has been used to graph 3

34

y x

and to create a table of values for this equation. Use the graph and the table to determine the x- and y-intercepts.

x -intercept: __________

y -intercept: __________

Graph

1,5,1 by 1,4,1

Table

A common expression is “Two points determine a ____________ ____________.” To obtain the graph of a linear equation, we can find any two points that satisfy the equation and then draw a line through these points.

11. Consider the equation 2 4y x (a) Select 3 input values and find their corresponding output values. List these ordered pairs in the table and plot them on the graph. The line through these points is the graph of

Table:x y

x-intercept:

____________

y-intercept:

____________

Graph:

2 4y x .

-1

0

1

2

3

4

5

6

7

8

9

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

(b) Determine the x- and y-intercepts for this graph.

11. Consider the equation 2 4y x

Table:x y

x-intercept:

____________

y-intercept:

____________

Graph:

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y

x

(c) Then graph the line on your calculator and use the TRACE feature to check your results. Use the same window as shown below. (See Calculator Perspective 2.3.1)

12. If a car gets 30 miles per gallon and travels 60 miles per hour, then the car uses 2 gallons of gas every hour. Assuming the car’s gas tank starts with 15 gallons, write a function that represents the number of gallons of gas remaining after x hours. ________ _______f x x

(a) Use a calculator to graph this equation using a window of 0,10,1 by 0,15,5

Draw a rough sketch of your calculator graph below.

0

5

10

15

0 1 2 3 4 5 6 7 8 9 10

y

x

(b) Press TRACE. Type 0 and ENTER. The y-intercept is ________________.

(c) Meaning of y-intercept:

(d) Then type 7.5 and ENTER. The x-intercept is ________________.

(e) Meaning of y-intercept:

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.

13. (a) Check each point to test if it is a solution of each equation:

Point Solution of ?

(yes/no)

Solution of ?

(yes/no)

7y x 3 5y x

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?

14. Consider the graph of the system: 24

3

xy

y x

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

y

x

(a) The solution of the system of equations shown is __________.

(c) Use the Intersect feature on your calculator to check this solution. (See Calculator Perspective 2.3.2 for help.)

(b) Verify that this point satisfies both equations.

15. Consider the system 3 5

7

y x

y x

.

Use the table of values to determine the solution of this system of equations.

Solution:

__________________

2 ______ 50y x

16. Suppose you go to rent a car and have two rental companies to choose from. The first company offers a rate of $40 per day plus 10 cents per mile, and the second company offers a rate of $25 per day plus 25 cents per mile. The cost for a two-day rental would be determined by the following equations:

Company A: 1 0.10 ______y x

Company B:

(a) Use your calculator to graph each equation using a window of 0,500,100 by 0,200,50

(b) Use a calculator to find the solution of the system. ____________

(c) Interpret this solution.

0

50

100

150

200

0 100 200 300 400 500

y

x

Draw a rough sketch of your calculator graph below.