Section 4.2: Review
Name: Period:
Directions: Solve the following systems of equations algebraically. Show all of your work.
1. 3 − −
(−4 5 −5)
2. 5 1
− 1 5
7 1 5
3. 3 4 1 − 4
(5 9 8)
4. − 8
5 1
(−1 17)
5. −6 − 3
5
infinitely many solutions
6. −
− 1
( 1 5 4)
8. Explain in words how you know when a system of linear equations has no solution. Justify your reasoning with an example of equations.
9. Explain in words how you know when a system of linear equations has infinitely many solutions. Justify your reasoning with an example of equations.
10. A system of equations is shown below. State the equations in slope-intercept form. State the solution as an ordered pair. Verify the solution using elimination.
11. The solution of a system of equations is −3 1 . On the coordinate plane below, graph two lines that could be the graphs of two linear equations in this system (do not use horizontal or vertical lines). Label your lines in slope-intercept form. Justify your equations by solving your system with substitution.
Equation #1:
Equation #2:
System solved by Substitution: Solution: (2,6)
Verify solution by elimination: Equation #1: y = 2x +2
12. Find the point on the graph of 4 − 5 −1 where the y-value is three more than three-fifths the x-value. Solve algebraically and show all of your work.
(5,6)
13. Solve the system of linear equations. Show all of your work. 3 −1
−3 − 3
(-1,0,1)
Directions: Define variables and create a system of equations to solve each word problem. Show all of your work. Write your final answer in a complete sentence.
14. A video rental company offers a plan that includes a membership fee of $6 and charges $5 for every DVD borrowed. They also offer a second plan that costs $31 per month for unlimited DVD rentals. How many DVD’s should a customer borrow so that each plan costs the same amount?
Let x = number of DVD’s borrowed Let y = total cost of either plan 5 6
31
15. At Card Crazy, the cost of five protective card sleeves and three Yu-Gi-Oh cards is $8.10. The cost of three protective card sleeves and four Yu-Gi-Oh cards is $6.40. If each protective sleeve costs the same amount and each Yu-Gi-Oh card costs the same amount, what is the price for one protective card sleeve? What is the price for one Yu-Gi-Oh card?
Let x = cost of one protective card sleeve Let y = cost of one Yu-Gi-Oh card
5 3 8 1
3 4 6 4
The cost of one protective card sleeve is $1.20 and the cost of one Yu-Gi-Oh card is $0.70.
16. Aria has a total of $3.95 in her pocket, all in quarters and dimes. She has 23 coins all together. How many quarters does she have? How many dimes does she have?
Let q = number of quarters Let d = number of dimes
3
0. 5 1 3 95
Aria has 11 quarters and 12 dimes.
17. A boat travels 486 miles in 9 hours going upstream and 470 miles in 5 hours going downstream. What is the speed of the boat in still water? What is the speed of the current?
9 − 486 → − 54 5 47 → 94
18. A farmer wants to mix milk containing 3% buttermilk with milk containing 30% buttermilk to obtain 900 gallons of milk which is 12% buttermilk. How much of each must he use?
9
3 3 1 9
The farmer must use 600 gallons of 3% buttermilk and 300 gallons of 30% buttermilk.
19. Graph the system of linear inequalities on the coordinate plane below. Label your lines. Shade the feasible region of solutions.
20. Young adults between the ages of 11 and 18 should get at least 1200 milligrams of calcium each day. One ounce of mozzarella cheese has 147 milligrams of calcium, and one ounce of Swiss cheese has 219 milligrams. If you wanted to eat no more than 8 ounces of cheese, how much of each type could you eat and still get your daily requirement of calcium?
b. Use your variables to write inequalities to describe the constraints of this situation. Explain the constraint that each of your inequalities represents.
147 19 1 Young adults should get at least 1200 milligrams of calcium each day 8 They want to eat at most 8 ounces of cheese
c. Graph your inequalities on the coordinate plane below. Label each axis and all of your lines. Shade the feasible region that represents the possible solutions. List three possible options.
Three possible solutions: answers will vary
examples:
3 ounces of mozzarella and 4 ounces of Swiss cheese
4 ounces of mozzarella and 3 ounces of Swiss cheese
5 ounces of mozzarella and 2.5 ounces of Swiss cheese
ou
n
ces of s
wi
ss
ch
ee
se
ounces of mozzarella cheese 𝑥 𝑦 8
