Objective:

• To solve systems of equations by graphing.

Black Friday Shopping

You want to buy some chocolate candy for your math teacher (ehem). The first website you find (chocolateisamazing.com) charges $3 plus $1 per pound to ship a box. The second website (hersheycandyisthebest.com) charges $1 plus $2 per pound to ship the same item.

For an object that weighs x pounds, the charges for the two websites are represented by the equations y = x + 3 and y = 2x + 1.

At what point are the charges the same?

Black Friday Shopping

• Create a table of values.

• Graph the equations.

• At what point do the two lines intersect? What does this ordered pair represent?

System of Equations:

• Two or more equations with the same set of variables are called a system of equations.

• A solution of a system of equations is an ordered pair that satisfies each equation in the system.

7 = 7

SOLUTION

EXAMPLE 1 Check the intersection point

Use the graph to solve the system. Then check your solution algebraically.

x + 2y = 7 Equation 1

3x – 2y = 5 Equation 2

The lines appear to intersect at the point (3, 2).

CHECK Substitute 3 for x and 2 for y in each equation.x + 2y = 7

3 + 2(2)=?

7

ANSWER

Because the ordered pair (3, 2) is a solution of each equation, it is a solution of the system.

EXAMPLE 1 Check the intersection point

3x – 2y = 5

5 = 53(3) – 2(2) 5=

?

EXAMPLE 2 Use the graph-and-check method

Solve the linear system:

–x + y = –7 Equation 1

x + 4y = –8 Equation 2

SOLUTION

STEP 1

Graph both equations.

EXAMPLE 2

STEP 2

Use the graph-and-check method

Estimate the point of intersection. The two lines appear to intersect at (4, – 3).

STEP 3

Check whether (4, –3) is a solution by substituting 4 for x and –3 for y in each of the original equations.

Equation 1–x + y = –7

–7 = –7–(4) + (–3) –7=

?

Equation 2x + 4y = –8

–8 = –84 + 4(–3) –8=

?

ANSWER

Because (4, –3) is a solution of each equation, it is a solution of the linear system.

EXAMPLE 2 Use the graph-and-check method

EXAMPLE 2 Use the graph-and-check method

Solve the linear system by graphing. Check your solution.

GUIDED PRACTICE for Examples 1 and 2

–5x + y = 01.5x + y = 10

ANSWER

(1, 5)

EXAMPLE 2 Use the graph-and-check method

Solve the linear system by graphing. Check your solution.

GUIDED PRACTICE for Examples 1 and 2

2x + y = 4–x + 2y = 32.

ANSWER

(1, 2)

EXAMPLE 2 Use the graph-and-check method

Solve the linear system by graphing. Check your solution.

GUIDED PRACTICE for Examples 1 and 2

3x + y = 3x – y = 53.

ANSWER

(2, 3)

EXAMPLE 3 Standardized Test Practice

As a season pass holder, you pay $4 per session to use the town’s tennis courts.

•

• Without the season pass, you pay $13 per session to use the tennis courts.

The parks and recreation department in your town offers a season pass for $90.

GUIDED PRACTICE for Example 3

4. Solve the linear system in Example 3 to find the number of sessions after which the total cost with a season pass, including the cost of the pass, is the same as the total cost without a season pass.

ANSWER 10 sessions

EXAMPLE 4 Solve a multi-step problem

A business rents in-line skates and bicycles. During one day, the business has a total of 25 rentals and collects $450 for the rentals. Find the number of pairs of skates rented and the number of bicycles rented.

RENTAL BUSINESS

EXAMPLE 4 Solve a multi-step problem

STEP 3

Estimate the point of intersection. The two lines appear to intersect at (20, 5).

STEP 4Check whether (20, 5) is a solution.

20 + 5 25=? 15(20) + 30(5) 450=?

450 = 45025 = 25

ANSWER

The business rented 20 pairs of skates and 5 bicycles.