Algebra practice questions
Linear equations, inequalities, systems, and proportional relationships. Most items reward a clean first step: isolate, substitute, or set the two expressions equal before doing any arithmetic.
Algebra questions
- 1Linear equationsFoundation
If 3x + 5 = 23, what is the value of x?
- A6
- B4
- C28
- D8
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Answer: 6
Subtract 5 from both sides to get 3x = 18. Divide by 3, so x = 6.
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For the line y = 2x - 3, what is the value of y when x = 4?
- A8
- B2
- C11
- D5
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Answer: 5
Substitute 4 for x: y = 2(4) - 3 = 8 - 3 = 5.
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If x + y = 10 and x - y = 4, what is the value of x?
- A5
- B3
- C14
- D7
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Answer: 7
Add the equations: 2x = 14. Therefore x = 7.
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What is the solution to 5(x - 2) = 3x + 10?
- A5
- B15
- C20
- D10
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Answer: 10
Expand first: 5x - 10 = 3x + 10. Subtract 3x and add 10: 2x = 20, so x = 10.
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If f(x) = 4x + 1 and f(t) = 21, what is the value of t?
- A20
- B5
- C4
- D6
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Answer: 5
Set 4t + 1 equal to 21. Then 4t = 20, so t = 5.
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What is the value of x in (x + 3) / 4 = 5?
- A7
- B17
- C20
- D23
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Answer: 17
Multiply both sides by 4: x + 3 = 20. Subtract 3 to get x = 17.
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If 0.4x = 18, what is x?
- A7.2
- B22.5
- C45
- D72
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Answer: 45
Divide 18 by 0.4. Since 18 / 0.4 = 45, x = 45.
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Which inequality is equivalent to 2x - 7 > 5?
- Ax < 1
- Bx < 6
- Cx > 1
- Dx > 6
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Answer: x > 6
Add 7 to both sides to get 2x > 12. Divide by positive 2, so x > 6.
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In a direct variation, y = kx. If y = 12 when x = 3, what is y when x = 7?
- A21
- B16
- C84
- D28
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Answer: 28
First find k: 12 = 3k, so k = 4. Then y = 4(7) = 28.
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What is the solution to 4 - 2x = 10?
- A3
- B-3
- C2
- D-2
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Answer: -3
Subtract 4 from both sides: -2x = 6. Divide by -2, so x = -3.
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Which expression is equivalent to 3(x + 2) - 2?
- A3x + 4
- B3x - 4
- Cx + 4
- D3x + 6
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Answer: 3x + 4
Distribute 3: 3x + 6 - 2 = 3x + 4.
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What is the slope of the line passing through (1, 4) and (3, 10)?
- A2
- B6
- C4
- D3
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Answer: 3
Slope is the change in y over the change in x: (10 - 4) / (3 - 1) = 6 / 2 = 3.
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What is the solution to (2x) / 3 - 1 = 5?
- A9
- B12
- C8
- D6
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Answer: 9
Add 1 to both sides to get 2x / 3 = 6. Multiply by 3 to get 2x = 18, so x = 9.
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If y = 3x - 2 and 2x + y = 13, what is the value of x?
- A5
- B4
- C3
- D2
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Answer: 3
Substitute 3x - 2 for y: 2x + 3x - 2 = 13, so 5x = 15 and x = 3.
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Which inequality is equivalent to -3x + 4 <= 19?
- Ax >= -5
- Bx <= 5
- Cx <= -5
- Dx >= 5
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Answer: x >= -5
Subtract 4 to get -3x <= 15. Dividing by a negative number reverses the inequality, so x >= -5.
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A line passes through (0, 7) and has slope -2. What is the value of y when x = 4?
- A1
- B15
- C-1
- D-15
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Answer: -1
The line is y = -2x + 7. Substituting x = 4 gives y = -8 + 7 = -1.
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If f(x) = 2x - 5, what is the value of f(f(3))?
- A-1
- B3
- C-3
- D1
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Answer: -3
First f(3) = 6 - 5 = 1. Then f(1) = 2 - 5 = -3.
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A recipe uses 5 cups of flour to make 2 loaves of bread. At the same rate, enter the number of cups needed for 7 loaves.
Student-produced response: write your own answer rather than choosing one.
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Answer: 17.5
The rate is 5 / 2 = 2.5 cups per loaf. For 7 loaves, 2.5 x 7 = 17.5 cups.
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If 3a + 2b = 12, which expression gives a in terms of b?
- A(12 + 2b) / 3
- B12 - 2b
- C(12 - 2b) / 3
- D(2b - 12) / 3
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Answer: (12 - 2b) / 3
Subtract 2b from both sides to get 3a = 12 - 2b, then divide both sides by 3.
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For which value of k does the system 2x + ky = 8 and 4x + 6y = 5 have no solution?
- A2
- B12
- C3
- D6
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Answer: 3
A system has no solution when the two lines are parallel but not identical. Multiplying the first equation by 2 gives 4x + 2ky = 16, which matches 4x + 6y in the x and y terms when 2k = 6, so k = 3. The constants 16 and 5 differ, so the lines never meet.
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What is the solution to 2(3x - 1) = 4x + 8?
- A5
- B3
- C4
- D6
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Answer: 5
Distribute to get 6x - 2 = 4x + 8. Subtract 4x and add 2 to get 2x = 10, so x = 5.
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The cost C, in dollars, of renting a bicycle for h hours is C = 4 + 3h. Enter the number of hours that gives a cost of 25 dollars.
Student-produced response: write your own answer rather than choosing one.
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Answer: 7
Set 4 + 3h = 25. Subtract 4 to get 3h = 21, so h = 7 hours.
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A line has slope 4 and passes through (2, 3). Which equation describes the line?
- Ay = 4x + 3
- By - 3 = 4(x - 2)
- Cy + 3 = 4(x + 2)
- Dy - 2 = 4(x - 3)
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Answer: y - 3 = 4(x - 2)
Point-slope form is y - y1 = m(x - x1). Substituting m = 4 and the point (2, 3) gives y - 3 = 4(x - 2). Note that the point's coordinates are subtracted, so both signs are negative here.
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A line is perpendicular to y = -2x + 5 and passes through the origin. What is its equation?
- Ay = (1/2)x
- By = 2x
- Cy = -2x
- Dy = -(1/2)x
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Answer: y = (1/2)x
Perpendicular slopes are negative reciprocals: the negative reciprocal of -2 is 1/2. A line through the origin has no constant term, so y = (1/2)x.
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How many values of x satisfy the equation |x - 4| = 7?
- A2
- BInfinitely many
- C0
- D1
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Answer: 2
An absolute value equals 7 when the expression inside is 7 or -7. Solving x - 4 = 7 gives x = 11 and x - 4 = -7 gives x = -3, so there are two solutions.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 26Interpreting linear modelsFoundation
A phone plan costs C = 15 + 0.05m dollars, where m is the number of minutes used. What does the 0.05 represent?
- AThe total cost of the plan
- BThe number of free minutes
- CThe fixed monthly fee
- DThe cost of each additional minute
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Answer: The cost of each additional minute
In a linear model, the coefficient of the variable is the rate of change. Here each extra minute adds $0.05, while the 15 is the fixed fee that applies regardless of usage.
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If 3x + 2y = 16 and 3x - 2y = 4, what is the value of y?
- A6
- B3
- C4
- D2
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Answer: 3
Subtract the second equation from the first: (3x + 2y) - (3x - 2y) = 16 - 4 gives 4y = 12, so y = 3. Elimination is cheaper here than substitution because the x terms already match.
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A delivery van can carry at most 900 kilograms. It is already loaded with 240 kilograms. If each crate weighs 30 kilograms, what is the greatest number of crates it can still take?
- A30
- B23
- C20
- D22
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Answer: 22
The remaining capacity is 900 - 240 = 660 kilograms. Dividing by 30 gives 22 crates exactly, and 23 crates would exceed the limit.
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If f(x) = 5 - 2x, for what value of x does f(x) = f(1) + 6?
- A2
- B-3
- C-2
- D4
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Answer: -2
First f(1) = 5 - 2 = 3, so the target value is 3 + 6 = 9. Solving 5 - 2x = 9 gives -2x = 4 and x = -2.
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What is the solution to 7 - 3(x - 1) = 1?
- A3
- B2
- C1
- D-3
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Answer: 3
Distribute carefully: 7 - 3x + 3 = 1, so 10 - 3x = 1. Then -3x = -9 and x = 3. Writing 7 - 3x - 3 is the common slip.
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If y varies directly with x and y = 18 when x = 4, enter the value of x when y = 45.
Student-produced response: write your own answer rather than choosing one.
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Answer: 10
Direct variation means y = kx with a constant k. From 18 = 4k, k = 4.5. Then 45 = 4.5x gives x = 10.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 32Interpreting linear modelsStandard
The number of members of a club is modelled by M = 48 + 7t, where t is years since opening. What was the membership at opening, and how fast is it growing?
- A7 members, growing by 48 a year
- B48 members, growing by 7 a year
- C48 members, growing by 48 a year
- D55 members, growing by 7 a year
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Answer: 48 members, growing by 7 a year
At t = 0 the model gives M = 48, which is the starting value. The coefficient of t is the annual rate of change, so membership grows by 7 each year.
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If (3x - 1) / 2 = (x + 7) / 3, what is the value of x?
- A5
- B17/7
- C17/11
- D3
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Answer: 17/7
Cross-multiply: 3(3x - 1) = 2(x + 7), so 9x - 3 = 2x + 14. Then 7x = 17 and x = 17/7.
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Which values of x satisfy both 2x + 1 > 7 and 5 - x > 0?
- AThere are no such values.
- B3 < x < 5
- Cx < 5
- Dx > 3
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Answer: 3 < x < 5
The first gives 2x > 6, so x > 3. The second gives x < 5. Both hold only between them, which is 3 < x < 5.
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The system 3x + ky = 9 and 6x + 10y = 18 has infinitely many solutions. What is k?
- A10
- B5
- C3
- D20
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Answer: 5
Infinitely many solutions means the equations describe the same line. Doubling the first gives 6x + 2ky = 18, which matches the second when 2k = 10, so k = 5. The constants already agree, which is what makes them identical rather than parallel.
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If f(x) = 3x + 2 and g(x) = x - 4, what is f(g(5))?
- A5
- B11
- C13
- D17
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Answer: 5
Work from the inside out: g(5) = 1, then f(1) = 3 + 2 = 5. Applying f first would give a different and incorrect answer.
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A machine fills 250 bottles in 20 minutes at a constant rate. Enter the number of bottles it fills in 50 minutes.
Student-produced response: write your own answer rather than choosing one.
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Answer: 625
The rate is 250 / 20 = 12.5 bottles per minute, and 12.5 x 50 = 625 bottles.
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Which expression is equivalent to 2(x - 3) - 3(2x + 1)?
- A-4x - 9
- B-4x + 9
- C8x - 9
- D-4x - 3
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Answer: -4x - 9
Distribute both: 2x - 6 - 6x - 3. Note the -3 multiplies both terms in the second bracket, giving -3 rather than +3. Collecting gives -4x - 9.
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A line passes through (-2, 7) and (4, -5). At what value of x does it cross the x-axis?
- A3.0
- B1.5
- C-1.5
- D2.0
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Answer: 1.5
The slope is (-5 - 7) / (4 - (-2)) = -12 / 6 = -2. Using (4, -5): -5 = -2(4) + b gives b = 3, so y = -2x + 3. Setting y = 0 gives x = 1.5.
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Which inequality describes all values within 4 units of 9 on the number line?
- A|x - 4| <= 9
- B|9 - 4| <= x
- C|x + 9| <= 4
- D|x - 9| <= 4
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Answer: |x - 9| <= 4
Distance from 9 is written |x - 9|, and within 4 units means that distance is at most 4. The subtraction names the centre, not the radius.
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If 8 - 2x = 3x - 7, what is the value of x?
- A-3
- B15
- C3
- D1
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Answer: 3
Add 2x to both sides: 8 = 5x - 7. Add 7: 15 = 5x, so x = 3.
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If 2x + 5y = 26 and x = y + 1, what is the value of y?
- A26/7
- B3
- C24/7
- D4
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Answer: 24/7
Substitute x = y + 1: 2(y + 1) + 5y = 26, so 7y + 2 = 26 and 7y = 24, giving y = 24/7.
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A taxi charges a $3 flag fee plus $1.80 per mile. What is the greatest whole number of miles that can be travelled for $30?
- A15
- B17
- C14
- D16
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Answer: 15
Solve 3 + 1.80m <= 30, so 1.80m <= 27 and m <= 15. Exactly 15 miles costs $30, and 16 would exceed it.
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If f(x) = ax + 3 and f(2) = 11, what is f(5)?
- A26
- B20
- C19
- D23
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Answer: 23
From f(2) = 11: 2a + 3 = 11, so a = 4 and f(x) = 4x + 3. Then f(5) = 20 + 3 = 23.
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If 5 workers build a wall in 12 days, enter how many days 4 workers would take at the same rate.
Student-produced response: write your own answer rather than choosing one.
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Answer: 15
The total work is 5 x 12 = 60 worker-days. Dividing by 4 workers gives 15 days. Fewer workers means more days, so the numbers move in opposite directions.
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Which expression is equivalent to (6x^2 + 9x) / 3x, for x not equal to 0?
- A6x + 3
- B2x^2 + 3
- C2x + 9
- D2x + 3
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Answer: 2x + 3
Divide each term by 3x: 6x^2 / 3x = 2x and 9x / 3x = 3. Dividing only the first term is the common slip.
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Two lines are perpendicular. One has equation y = (3/4)x - 2. What is the slope of the other?
- A-3/4
- B-4/3
- C3/4
- D4/3
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Answer: -4/3
Perpendicular slopes are negative reciprocals: flip 3/4 to 4/3 and change the sign, giving -4/3.
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How many values of x satisfy |2x - 6| = -4?
- A1
- BInfinitely many
- C0
- D2
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Answer: 0
An absolute value is never negative, so no value of x can make it equal -4. Solving the two cases anyway produces answers that fail when checked.
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If 4(x + 2) - 3 = 2(x + 9), what is the value of x?
- A5
- B8
- C6.5
- D3.5
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Answer: 6.5
Distribute both sides: 4x + 8 - 3 = 2x + 18, so 4x + 5 = 2x + 18. Then 2x = 13 and x = 6.5.
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If 4x - y = 11 and x + y = 4, what is the value of x?
- A5
- B3
- C2
- D1
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Answer: 3
Adding the equations eliminates y: 5x = 15, so x = 3. Elimination is cheaper here than substitution because the y terms already cancel.
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A line has x-intercept 4 and y-intercept -6. What is its slope?
- A2/3
- B-2/3
- C-1.5
- D1.5
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Answer: 1.5
The line passes through (4, 0) and (0, -6). Slope is (0 - (-6)) / (4 - 0) = 6 / 4 = 1.5.
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For how many integers x is -2 < 3x - 5 <= 7?
- A3
- B5
- C4
- D2
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Answer: 3
Add 5 throughout: 3 < 3x <= 12. Divide by 3: 1 < x <= 4. The integers are 2, 3 and 4, so there are three.
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If f(x) = x^2 + 1, which value of x gives f(x) = f(-3)?
- A10
- B3
- C9
- D-9
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Answer: 3
f(-3) = 10, and f(3) = 10 as well because squaring removes the sign. The value 10 is the output, not the input.
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On a map, 3 centimetres represents 12 kilometres. Enter the number of kilometres represented by 7.5 centimetres.
Student-produced response: write your own answer rather than choosing one.
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Answer: 30
The scale is 12 / 3 = 4 kilometres per centimetre, and 4 x 7.5 = 30 kilometres.
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If a = 3b and b = 2c, which expression gives a in terms of c?
- A6c
- Bc/6
- C1.5c
- D5c
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Answer: 6c
Substitute b = 2c into a = 3b: a = 3(2c) = 6c. Adding the coefficients rather than multiplying gives 5c, which is the trap.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 56Interpreting linear modelsStandard
A tank drains according to V = 500 - 25t, where V is litres and t is minutes. How long does it take to empty?
- A475 minutes
- B25 minutes
- C20 minutes
- D500 minutes
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Answer: 20 minutes
Empty means V = 0, so 25t = 500 and t = 20 minutes. The 25 is the rate, not the time.
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