Topic 6 Review

- List the methods to solve systems of linear equations. - Describe the possible graphical solutions to a system of equations. List the methods to solve systems of linear equations. (substitution, elimination, or graphically) Describe the possible graphical solutions to a system of equations.

1- Solve the following system of equations: 𝑥−2𝑦=3 3𝑥+𝑦=−5 Solve for x Solve by Elimination: 𝑥−2𝑦=3 2(3𝑥+𝑦=−5) Solve by Substitution: 𝑥=3+2𝑦 3(3+2𝑦)+𝑦=−5 Substitute 𝑥−2𝑦=3 6𝑥+2𝑦=−10 9+6𝑦+𝑦=−5 9+7𝑦=−5 7𝑦=−14 7𝑥=−7 x=−1 𝑦=− 14 7 𝑦=−2 x−2(−2)=3 𝑥+4=3 𝑥=−1 −1 −2𝑦=3 −2𝑦=3+1 −2𝑦=4 𝑦=−2 The solution for this system of equations is (-1,-2)

2- Solve the following system of equations: 2𝑥+3𝑦=4 4𝑥+6𝑦=9 Solve by Elimination: −2(2𝑥+3𝑦=4) 4𝑥+6𝑦=9 −4𝑥−6𝑦=−8 4𝑥+6𝑦=9 0=1 This system of equations has no solutions

3- Solve the following system of equations: 𝑥=2𝑦−4 −4𝑥+8𝑦=16 Substitute Solve by Substitution: −4 2𝑦−4 +8𝑦=16 −8𝑦+16+8𝑦=16 16=16 This system of equations infinitely many solutions

4- Solve the following system of equations: 11−5𝑦=2𝑥 5𝑦+3=−9𝑥 Solve by Elimination: 11−5𝑦=2𝑥 3+5𝑦=−9𝑥 Rearrange the “y” 14=−7𝑥 𝑥= 14 −7 𝑥=−2 11−5𝑦=2 −2 11−5𝑦=−4 −5𝑦=−4−11 −5𝑦=−15 𝑦=− 15 −5 𝑦=3 This system of equations a solution at (-2,3)

First establish variables: r = Roast Beef , m = Mashed Potatoes 5- Roast beef has 25g of protein and 11g of calcium per serving. A serving of mashed potatoes has 2g of protein and 25g of calcium. How many servings of each are needed to exactly supply 29g of protein and 61g of calcium? Then, set up the equations with the information given. 25𝑟+2𝑚=29 11𝑟+25𝑚=61 Solve by substitution 25 1 +2𝑚=29 25𝑟+2𝑚=29 25+2𝑚=29 2𝑚=29−25𝑟 2𝑚=4 𝑚= 29−25𝑟 2 =14.5−12.5𝑟 𝑚=2 Add r = roast… P = potatoes 11𝑟+25(14.5−12.5𝑟)=61 You will need exactly one serving of Roast Beef and two servings of mashed potatoes. 11𝑟+362.5−312.5𝑟=61 362.5−301.5𝑟=61 −301.5𝑟=−301.5 𝑟=1

6- You want to get a sales job to be able to purchase your first car 6- You want to get a sales job to be able to purchase your first car. Company A offers you $200 a week plus $10 for each item you sell. Company B offers you $250 a week but only $5 for each item you sell. Let “x” be the number of items you sell in a week, and “y” be the amount of money you made. How many items must you sell before the amount of money made is the same for both companies? First, set up the equations with the information given. Company A Company B 𝑦=10𝑥+200 𝑦=5𝑥+250 Let’s substitute either one of the “y” 10𝑥+200=5𝑥+250 5𝑥=50 𝑥=10

7- Solve the following system of equations: 𝑦=5𝑥+11 𝑦=−2 𝑥+1 2 +4 Let’s substitute either one of the “y” −2 𝑥+1 2 +4=5𝑥+11 FOIL −2( 𝑥 2 +2𝑥+1)+4=5𝑥+11 Distribute the -2 multiplying the parenthesis −2 𝑥 2 −4𝑥−2+4=5𝑥+11 Combine like-terms −2 𝑥 2 −4𝑥+2=5𝑥+11 To solve a quadratic expression it has to be “= 0” −2 𝑥 2 −4𝑥+2−5𝑥−11=0 Combine like-terms −2 𝑥 2 −9𝑥−9=0 Factor out the negative sign Leading coefficient, “a”, not 1. Then, multiply “a” by “c” and find factors of that value that add up to “b” −(2 𝑥 2 +9𝑥+9)=0 −(2 𝑥 2 +6𝑥+3𝑥+9)=0 −( 2 𝑥 2 +6𝑥 +(3𝑥+9))=0 𝑎 𝑥 2 +𝑏𝑥+𝑐 −(2𝑥 𝑥+3 +3(𝑥+3))=0 −(2𝑥+3)(𝑥+3)=0 2∗9=18 𝑥=−3/2 ; 𝑥=−3 Solve for “x” 18=3∗6 𝑦=−7/2 ;𝑦=−4 Find the corresponding values for “y” 3+6=9

8- Solve the following system of equations: 𝑥−𝑦=3 (𝑥−2) 2 + 𝑦+3 2 =4 Solve the first equation for “x”: 𝑥=3+𝑦 Substitute into second equation: 3+𝑦−2 2 + 𝑦+3 2 =4 𝑦+1 2 + 𝑦+3 2 =4 FOIL ( 𝑦 2 +2𝑦+1)+( 𝑦 2 +6𝑦+9)=4 Combine like-terms 2𝑦 2 +8𝑦+10=4 To solve a quadratic expression it has to be “= 0” 2 𝑦 2 +8𝑦+6=0 Factor out 2 2( 𝑦 2 +4𝑦+3)=0 Leading coefficient, “a”, is 1. So find factors of “c” that add up to “b” 2(𝑦+3)(𝑦+1)=0 𝑦=−3 ;𝑦=−1 Solve for “y” 𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑥=0 ;𝑥=2 Find the corresponding values for “x” 𝑐=3 3=3∗1 3+1=4

9- The FSA has a math and a reading section 9- The FSA has a math and a reading section. A student can score a maximum of 1000 points in each. To qualify, a student has to score at least 700 points in math and a minimum of 1200 total points. Write a system of inequalities to represent the possible combinations of math scores, x, and reading scores, y, of a student who passes.

9- The FSA has a math and a reading section 9- The FSA has a math and a reading section. A student can score a maximum of 1000 points in each. To qualify, a student has to score at least 700 points in math and a minimum of 1200 total points. Write a system of inequalities to represent the possible combinations of math scores, x, and reading scores, y, of a student who passes. Answer: 700≤𝑥≤1000 0≤𝑦≤1000 x+y≥1200

10- A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $125 for the rear-projection televisions and $200 for the plasma televisions Let x = the number of rear-projection television manufactured in a month and let y = the number of plasma televisions manufactured in a month. Write a function that models the total monthly profit, P. The manufacturer is bound by the following constrains: Equipment in the factory allows for making at most 450 rear-projection televisions in one month. Equipment in the factory allows for making at most 200 plasma televisions in one month The cost to the manufacturer per unit is $600 for the rear-projection televisions and $900 for the plasma televisions. Total monthly costs cannot exceed $360,000. Write a system of three inequalities that models these constraints.

11- A student earns $10 per hour for tutoring and $7 per hour as a teacher’s aide. Let x = the number of hours each week spent tutoring and let y = the number of hours each week spent as a teacher’s aide. Write a function that models total weekly earnings, W. The student is bound by the following constraints; To have enough time for studies, the student can work no more than 20 hours per week. The tutoring center requires that each tutor spend at least three hours per week tutoring. The tutoring center requires that each tutor spend no more than eight hours per week as a teacher’s aide. Write a system of three inequalities that models these constraints.