SAT Math Practice Test: Algebra
- Algebra
- 11 questions
- 15 minutes
These 11 questions cover the SAT Algebra domain: linear equations and inequalities, systems of equations and linear functions in context. When a word problem gives two data points, find the rate of change first; most of the work follows from it.
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Question 1
If 3(2x − 5) = 4x + 7, what is the value of x?
Correct answer: C
Distributing gives 6x − 15 = 4x + 7, so 2x = 22 and x = 11. The choice 22 is the value of 2x, and 6 comes from forgetting to multiply −5 by 3.
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Question 2
A courier charges a fixed fee plus a constant rate per mile. A 12-mile delivery costs $31, and a 20-mile delivery costs $43. Which equation gives the total cost C, in dollars, of a delivery of m miles?
Correct answer: C
The rate is (43 − 31) ÷ (20 − 12) = 1.5 dollars per mile, and 31 = 1.5(12) + b gives b = 13. The equation C = 2.5m + 1 matches the 12-mile cost but gives $51, not $43, for 20 miles.
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Question 3
3x + 5y = 41
5x + 3y = 31If (x, y) is the solution to the system of equations shown, what is the value of x + y?
Correct answer: D
Adding the two equations gives 8x + 8y = 72, so x + y = 9 (here x = 2 and y = 7). The choice 5 is y − x, which comes from subtracting the equations instead of adding them.
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Question 4
Which of the following values of x satisfies the inequality 2(x − 4) > 5x + 7?
Correct answer: A
Expanding gives 2x − 8 > 5x + 7, so −3x > 15. Dividing by −3 reverses the inequality: x < −5, and only −6 is less than −5. The value −5 itself does not work because the inequality is strict.
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Question 5
4x − ky = 9
2x + 3y = 5In the system of equations shown, k is a constant. For what value of k does the system have no solution?
Correct answer: A
Doubling the second equation gives 4x + 6y = 10. The system has no solution when the first equation is 4x + 6y = 9, a parallel line, so −k = 6 and k = −6. The choice 6 is a sign error.
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Question 6
The function f(t) = 850 − 35t gives the number of liters of water remaining in a tank t minutes after the tank begins to drain. What is the best interpretation of the number 35 in this context?
Correct answer: B
The number 35 is the slope: each time t increases by 1 minute, the amount of water drops by 35 liters. The starting amount is 850 liters, and the tank is empty after 850 ÷ 35 ≈ 24.3 minutes, not 35.
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Question 7
In the xy-plane, a line passes through the points (−2, 5) and (4, −7). The line crosses the y-axis at the point (0, b). What is the value of b?
Correct answer: B
The slope is (−7 − 5) ÷ (4 − (−2)) = −2, so 5 = −2(−2) + b and b = 1. Using a slope of 2 by mistake gives 9.
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Question 8
Line ℓ is perpendicular to the line with equation 3x − 4y = 8 and passes through the point (6, 1). Which equation defines line ℓ?
Correct answer: A
The given line can be written as y = 34x − 2, so a perpendicular line has slope −43. Then 1 = −43(6) + b = −8 + b, so b = 9. The equation with slope 34 is parallel to the given line, not perpendicular.
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Question 9
A theater sold 420 tickets to a performance and collected a total of $5,880. Adult tickets cost $16 each, and student tickets cost $11 each. How many student tickets were sold?
Correct answer: B
With a + s = 420 and 16a + 11s = 5,880, substitute a = 420 − s: 6,720 − 5s = 5,880, so s = 168. The choice 252 is the number of adult tickets.
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Question 10
A gym offers two ways to pay for fitness classes. Members pay a one-time fee of $85 plus $12.50 per class, and nonmembers pay $20 per class. What is the least number of classes for which the total cost for a member is less than the total cost for a nonmember?
Correct answer: D
Solve 85 + 12.5n < 20n: 85 < 7.5n, so n > 11.33. The least whole number that works is 12; at 11 classes a member pays $222.50 and a nonmember pays only $220.
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Question 11
a(x + 3) − 2x = 5x + b
In the equation shown, a and b are constants. If the equation has infinitely many solutions, what is the value of a + b?
Correct answer: C
The left side simplifies to (a − 2)x + 3a. Infinitely many solutions require a − 2 = 5 and 3a = b, so a = 7, b = 21 and a + b = 28. The choice 20 comes from ignoring the −2x term.
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