Solving Systems of Equations

Transcription

1 Solving Sstems of Equations Eample 1: Emil just graduated from college with a degree in computer science. She has two job offers. Kraz Komputers will pa her a base salar of $0,000 with a $500 raise each ear that she works there. Pear Pads will pa her a base salar of $35,000 with a $1,000 increase each ear that she works there. How man ears would Emil have to work at Pear Pads to make the same salar as she would if she were to work at Kraz Komputers? Let equal the number of ears that Emil works at a compan and let be her salar. Write an equation to represent the salar for each job. Kraz Komputers: 0, Pear Pads: 35,000 1,000 Use a graphing calculator or graphing software to graph the two equations. The ordered pair (10, 5,000) is the intersection of the two lines. In other words, after 10 ears Emil would make $5,000 at either job. Eample : The state fair is a popular field trip destination. This ear the senior class at Sunnside High and the senior class at Riverside High both planned trips there. The senior class at Sunnside High rented and filled 8 vans and 8 buses with 0 students. Riverside High rented and filled vans and 1 bus with 5 students. Ever van had the same number of students in it as did the buses. Find the number of students in each van and in each bus. Define the variables. Let represent the number of students per van and let represent the number of students per bus. Write an equation for Sunnside High and Riverside High. Sunnside High: Riverside High: 5

2 Use a graphing calculator or graphing software to graph each equation. The ordered pair (8, ) is the intersection of the two lines. Because is representing the number of students per van, there are 8 students per van and because is representing the number of students per bus, there are students per bus. NUMBER OF SOLUTIONS OF A LINEAR SYSTEM The relationship between the graph of a linear sstem and the sstem s number of solutions is described below. Eactl One Solution Infinitel Man Solutions No solution 3

3 Eample 3: Solve each sstem of equations graphicall and determine if there is one, infinitel man or no solution. A) B) C) A) B) C) Infinitel Man Solutions No Solution One Solution (, 3)

4 Practice Set 1: Identif the solution Identif what the solutions and means in the contet Savings in $ Total Membership Cost Number of cats Number of weeks No. of Eercise Classes Number of birds 5

5 Practice Set : 1. Austin and Alec are selling fruit for a school fundraiser. Customers can bu boes of oranges or boes of apples. Austin sold 3 boes of oranges and 1 boes of apples for a total of $03. Alec sold 11 boes of oranges and 11 boes of apples for a total of $0. Find the cost of each bo of oranges and each bo of apples.. Jason and Britton are driving to St. George. Jason got a 0 mile head start and drove an average of 5 mph the entire wa. Britton drove an average of 75 mph. After how man hours and miles will Britton catch up to Jason? Solve each sstem of equations graphicall and determine if there is one, infinitel man or no solution

6 Solving Sstems Algebraicall Eample : Solve the sstem using the substitution method The first equation is alread solved for the variable. Substitute 11 into the second equation for. Simplif the equation and solve for. Substitute the value ou found for into the first equation to solve for. (, 1) Write our answer as an ordered pair. Eample 5: Solve the sstem 1 8 using the elimination method and are inverses of one another. If ou were to add them the sum would be 0. Adding the two equations will eliminate the terms and ou can solve for. Solve for. 7

7 Substitute our value for into either equation and solve for., Write the solution as an ordered pair. Eample : Solve the sstem using the elimination method There are no additive inverses, so ou will need to multipl one or both equations b a number to create an additive inverse. Multipl the first equation b -5 and the second equation b Solve for , Write the solution as an ordered pair. Adding the two equations will eliminate the terms and ou can solve for. Substitute our value for into either equation and solve for. 8

8 Practice Set 3: Solve each sstem of equations algebraicall The school that Sam goes to is selling tickets to a pla. On the first da of ticket sales, the school sold 3 adult tickets and 1 student ticket for a total of $38. The school took in $5 on the second da b selling 3 adult tickets and student tickets. Find the price of an adult ticket and the price of a student ticket. 10. Power Fit gm charges a $75 membership fee and $3 for each fitness class. Flabb Fitness gm charges a $100 membership fee and $ for each fitness class. How man fitness classes do ou have to attend at Power Fit gm to pa the same amount that ou would at Flabb Fitness? 11. At Cars R Us ou can rent a car for $5 a da plus $0.30 per mile. At Larr s Limos ou can be driven around in a limo for $110 a da plus $0.15 per mile. How man miles would ou have to drive a car from Cars R Us to equal the same amount as being driven around in one of Larr s Limos? 1. A jewelr designer purchased 80 beads for a total of $39 to make a necklace. Some of the beads were sterling silver beads that cost $0.0 each and the rest were imitation pearl beads that cost $0.5 each. How man of each tpe did the designer bu? 9