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 = (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 = 5 3(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–x + y = –7 –7 = –7 –(4) + (–3) –7 = ? Equation 2 x + 4y = –8x + 4y = –8 –8 = – (–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 = x + 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 = 3 2. 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 = 3 x – y = 5 3. 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 4 Check whether (20, 5) is a solution = ? 15( 20 ) + 30(5) 450 = ? 450 = = 25 ANSWER The business rented 20 pairs of skates and 5 bicycles.