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Math 105 Practice Problems - Exam 2

Do all work on your own paper. Show your steps .

1. Solve the system of equations by graphing . State whether there is a unique solution, no
solution, or an infinite number of solutions . If the answer is unique, ex press it as an ordered
pair.

x + y = –7
4x – y = –8

2. Solve the system of equations by graphing. State whether there is a unique solution, no
solution, or an infinite number of solutions. If the answer is unique, express it as an ordered
pair.

y = x + 1
3y – 3x = 6

3. Solve the system of equations by substitution. State whether there is a unique solution, no
solution, or an infinite number of solutions. If the answer is unique, express it as an ordered
pair.

y = 2x – 13
4x + 9y = –7

4. Solve the system of equations by substitution. State whether there is a unique solution, no
solution, or an infinite number of solutions. If the answer is unique, express it as an ordered
pair.

4x = 2y + 6
y = 2x – 3

5. Solve the system of equations by the addition method . State whether there is a unique
solution, no solution, or an infinite number of solutions. If the answer is unique, express it as
an ordered pair.

2x – 5y = 23
3x + 5y = –28

6. Solve the system of equations by the addition method. State whether there is a unique
solution, no solution, or an infinite number of solutions. If the answer is unique, express it as
an ordered pair.

–3x – 5y = 7
5x + 4y = 10

7. During a public ice skating session, an ice skating arena charged an admission fee for each
child plus a rental fee for each pair of ice skates. Byron Carter paid the admission fees for his
5 nephews and rented 4 pairs of ice skates. He was charged $37.75. Corrina Prescott paid the
admission fees for her 4 grandchildren and rented 2 pairs of ice skates. She was charged
$26.00. What was the admission fee? What was the skate rental fee?

8. A solution of 48% pesticide is to be mixed with a solution of 34% pesticide to form 56 liters
of a 44% solution. How many liters of the 48% solution must be used?

9. A store is having a sale on sweaters. A crew neck sweater sells for $24 and a V-neck sweater
sells for $18. If they sold 38 sweaters in one day , and took in $786.00, how many of each
type of sweater were sold?

10. The sum of the measures of the angles of a triangle is 180 degrees. The largest angle
measures 100 less than the sum of the other two. The measure of the middle-sized angle is
the average of the other two. Find the measures of the three angles.

11. Alex is developing film in his darkroom when he realizes that he does not have the proper
developer. He needs a mixture that is 20% developer but he only has a mixture that is 60%
developer and another that is 10% developer. How much of the 10% developer and how
much of the 60% developer must be mixed to produce 8 pints of the mixture he needs?

12. How much water should be added to 20 milliliters of a 30% hydrochloric acid solution in
order to dilute it to a 10% strength HCl solution?

13. Steve Fletcher wants to de sign a walkway to his house using two different kinds of paving
bricks. He finds the job will require 380 bricks. The home improvement store sells two
kinds of paving bricks that will work together, a red brick variety which sells for $5.00 per
unit and a cement brick which sells for $3.00 per unit. How many of each type of brick
should he buy if he has budgeted $1500 for the project?

14. A plane flies 560 mi. in 1.75 hr. traveling with the wind. The return trip later against the
wind takes the plane 2 hr. Find the speed of the plane and the speed of the wind. Let x =
the speed of the plane and y = the speed of the wind.

15. Rachelle does tutoring and paper grading for the math department . One week she tutored
for 7 hours and graded for 8 hours and made $107.00. The fol lowing week she tutored for
10 hours and graded for 6 hours and made $123.00. How much does she make per hour
tutoring and how much per hour grading?

16. A theater has an orchestra section and a balcony section. At one performance, the sale
of 95 orchestra seats and 75 balcony seats generated a revenue of $4975. At a later
performance, the sale of 60 orchestra seats and 135 balcony seats brought in $5070.
Find the cost of an orchestra seat and the cost of a balcony seat.

17. Simplify and write without negative exponents : –3a–3b

18. Simplify and write without negative exponents: 3x–1y

19. Evaluate each expression:

20. Simplify and write without negative exponents:

21. Simplify and write without negative exponents:

22. Add:

23. Subtract:

24. Subtract: from

25. Perform the indicated operations:

26. Multiply :

27. Multiply:

28. Multiply: (4x – 3y)(4x + 3y)

29. Expand and simplify : (3a – 2b)2

30. Use long division to divide: 4x2 + 2x – 15 by 2x + 5

31. Divide:

32. Divide:

33. a. Find the greatest common factor of

b. Factor the greatest common factor from

34. a. Find the greatest common factor of

b. Factor the greatest common factor (with a negative coefficient) from

35. Factor completely. If an expression cannot be factored with rational numbers , write
“Prime.”

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