Solving Systems of Equations by Graphing Graph each line on the same coordinate plane.

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1 Notes Systems Solving Systems of Equations by Graphing Graph each line on the same coordinate plane y x 5. y 3x x y 3x y Do they intersect? Where? What is the solution? Do they intersect? Where? What is the solution? 3. 1 y x 3 x 4y 0 Do they intersect? Where? What is the solution?

2 Find the solution to the following system of linear equations by filling in the table: y x 4 y x 3 x Is a given ordered pair a solution to a system of equations?? To be a solution, the ordered pair must work in both equations. Plug in to find out!!! Show work to prove your answer. State Yes or No. 4) Is, 4 a solution to the system? y x y x 6 5) Is 3, 1 a solution to the system? y x 5 y x 1

3 Total Cost Application 6) The Belton City Parks Department offers a season pass for $45 to have access to the city s tennis courts. As a season pass holder, you pay $3 per session for using the city courts. Without a season pass, you pay $1 per session for use of the courts. Writing a System of Equations which system of equations can be used to find the number of sessions of tennis, x, after which the total cost, y, with a season pass, is the same as the total cost without a season pass? a) y 3x b) y 3x c) y 45 3x d) y 45 3x y 1x y 45 1x y 1x y 45 1x Making a Table Make a table of values that shows the total cost for a season pass holder and a non season pass holder after 1,, 3, 4, 5, and 6 tennis sessions. After how many sessions is the cost the same? Drawing a Graph use the table to graph the system of equations. Under what circumstances does it make sense to become a season pass holder? # of Sessions Cost for Season Pass Holders Cost for Non- Pass Holders # of Tennis Sessions

4 Solving Systems of Equations by Substitution The figure at the right shows the graphs of two lines. Use the figure to estimate the coordinates of the point that belongs to both lines. The system of equations is: y = x x + 5y = -1 Dale took one look at these equations and offered a plan: The first equation says you can substitute x for y in the second equation. Then you have only one equation to solve. Find the missing y value by inserting the x value you found into either of the two original equations. Do the coordinates of the intersection point agree with your estimate? 1. y = x + 1 3x + y = x + y = 6 x + y = 4. x = -4y + 1 x 3y = y = 3x -7 3x y = 7

5 Kinley is buying pens and pencils to start the new school year. She spent a total of $18. Each pencil costs $0.40 and each pen costs $1.00. She bought twice as many pencils as pens. How many of each did she buy? a) Define the variables. The band booster club is selling pompoms and spirit sticks to raise money. Pom- poms cost $3.50 each and spirit sticks cost $4.00 each. The booster club has made a total of $43.50 and has sold three times as many pom-poms as spirit sticks. How many of each did they sell? a) Define the variables. b) Write the system of equations that can be used to find how many of each she bought. b) Write the system of equations that can be used to find how many of each they sold. c) Use substitution to solve the system. c) Use substitution to solve the system. d) Answer the question in the problem appropriately. d) Answer the question in the problem appropriately.

6 Solving Systems of Equations by Elimination The system of equations that has been graphed is : x + 5y = -8 -x + 3y = -8 Jess took one look at these equations and knew right away what to do. Just add the equations and you will find out quickly what x is. Follow this advice, and explain why it works. 1. 4x + 3y = 0 5x 3y = 7 3. x 5y = 16 4x = -5y +. 3x y = 1-3x + y = y + 4 = 15x 15x + 8y = 4

7 The figure at the right shows the graphs of two lines. Use the figure to estimate the coordinates of the point that belongs to both lines. The system of equations is x + 3y = 5 5x + 4y = 16 Lee took one look at these equations and announced a plan: Just multiply the first equation by 5 and the second equation by -. What does changing the equations in this way do to their graphs? 5. 5x - 3y = -18 x 6y = x = 3y + 7 x + y = 6. = 4x - y 3 = -x + y 8. 5x y = -1 3x = 5y + 7

8 Systems of Equations Applications 1. There are 88 boats in the harbor. Some are sailboats; the rest are motor-powered. There are 9 more motor-powered boats than sailboats. How many sailboats are there? 4. An automobile dealer sold 180 vans and trucks at a sale. He sold 40 more vans than trucks. How many of each did he sell?. Joel has 14 coins, all dimes and quarters, worth $.60. How many dimes and quarters does Joel have? 5. Ruth is twice as old as Pat. In 4 years Ruth will be three times as old as Pat was 3 years ago. How old is each now? 3. The length of a rectangle is 3 cm more than twice the width. If the perimeter is 84 cm, find the dimensions. 6. Alex has $ more than Fred has. Altogether they have $9. How much money does Alex have?

9 7. Amanda bought roses for $3.0 a bundle and daisies for $1.50 a bundle. She bought a total of 4 bundles for $ How many bundles of each type of flower did Amanda buy? 8. Last year, you mowed grass and trimmed trees for 1 households. You earned $5 for mowing a household s lawn for the entire year and $00 for trimming their trees. You earned a total of $600 last year. How many households did you mow the lawn for and how many households did you trim trees? 9. Solve the system using any method: (graphing, substitution, or elimination) x y 6 3x 5y 33