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
- 57SlopeFoundation
A line passes through the points (2, 7) and (6, 19). What is its slope?
- A3
- B1/3
- C4
- D-3
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Answer: 3
Slope is the change in y over the change in x: (19 - 7) / (6 - 2) = 12 / 4 = 3. Dividing the other way round gives 1/3, which is the most common slip here.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 58Interpreting linear modelsStandard
The cost in dollars of hiring a floor sander for one day and running it for m minutes is C(m) = 45 + 0.12m. What does 45 represent?
- AThe cost of each minute of use
- BThe cost before the sander is used at all
- CThe number of minutes included in the hire
- DThe largest amount the hire can cost
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Answer: The cost before the sander is used at all
Setting m = 0 gives C(0) = 45, so 45 is what the hire costs before any running time is added. The 0.12 is the per-minute charge, and nothing in the model caps the total.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 59Interpreting linear modelsStandard
A phone plan costs P(g) = 18 + 6g dollars a month, where g is the number of extra gigabytes used. If a customer uses 3 more gigabytes than in a month with no extras, by how much does the bill rise?
- A$6
- B$54
- C$18
- D$24
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Answer: $18
Each extra gigabyte adds 6 dollars, so three add 3 x 6 = 18 dollars. The 18 in the model is the base charge, not the increase, and 54 is what the whole bill becomes rather than how much it rises.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 60Absolute valueStandard
What is the sum of all solutions to |2x - 5| = 9?
- A2
- B9
- C-2
- D5
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Answer: 5
The expression inside the bars is 9 or -9. From 2x - 5 = 9, x = 7. From 2x - 5 = -9, x = -2. Their sum is 7 + (-2) = 5. Solving only the first equation is what leaves you with a single answer.
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If 4x + 3y = 27 and 2x - y = 1, what is the value of x + y?
- A8
- B3
- C5
- D2
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Answer: 8
The second equation gives y = 2x - 1. Substituting: 4x + 3(2x - 1) = 27, so 10x = 30 and x = 3, making y = 5. Then x + y = 8. The wrong answers are x and y themselves, which is what you get by stopping one step early.
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A van can carry at most 1,200 kg. It already holds 340 kg of equipment, and each box loaded weighs 24 kg. What is the greatest number of boxes it can take?
- A36
- B35
- C50
- D64
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Answer: 35
The condition is 340 + 24b <= 1200, so 24b <= 860 and b <= 35.83. Because b counts boxes it must be a whole number, and rounding up to 36 puts the load at 1,204 kg, over the limit.
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The function f is defined by f(x) = 5x - 4. If f(a) = 26, what is the value of f(a + 2)?
- A28
- B30
- C36
- D52
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Answer: 36
From 5a - 4 = 26 comes a = 6, so f(8) = 40 - 4 = 36. Quicker: adding 2 to the input of a linear function with slope 5 adds 10 to the output, and 26 + 10 = 36. Doubling the output gives 52, which the notation does not license.
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Which expression is equivalent to (3x + 2)(x - 5) - 3x^2?
- A13x - 10
- B-13x + 10
- C6x^2 - 13x - 10
- D-13x - 10
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Answer: -13x - 10
Expanding the product gives 3x^2 - 15x + 2x - 10 = 3x^2 - 13x - 10. Subtracting 3x^2 cancels the squared term and leaves -13x - 10. Forgetting to subtract leaves the quadratic term in place; dropping a sign flips the linear term.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 65Direct variationFoundation
The cost of fencing varies directly with its length. If 12 metres costs 90 pounds, what do 20 metres cost?
- A150 pounds
- B98 pounds
- C135 pounds
- D54 pounds
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Answer: 150 pounds
Direct variation means cost = k x length. From 90 = 12k comes k = 7.5 pounds per metre, so 20 metres cost 150 pounds. Adding 8 pounds for the 8 extra metres gives 98, which treats a rate as a fixed sum.
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The distance a spring stretches varies directly with the force applied. A force of 8 newtons stretches it 5 cm. Enter the stretch, in centimetres, produced by a force of 20 newtons.
Student-produced response: write your own answer rather than choosing one.
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Answer: 12.5
The constant is 5 / 8 = 0.625 cm per newton, so 20 newtons give 20 x 0.625 = 12.5 cm. Setting up the proportion 5/8 = x/20 and cross-multiplying reaches the same place.
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A line has slope -2/3 and passes through the point (6, 1). What is its y-intercept?
- A-3
- B5
- C1
- D9
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Answer: 5
Substituting into y = mx + b gives 1 = (-2/3)(6) + b = -4 + b, so b = 5. The choice 1 is the y-coordinate of the given point, which is only the intercept when x is 0.
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Line k passes through (-2, 5) and (4, -7). Which line is parallel to line k?
- Ay = 2x - 1
- By = -1/2 x + 3
- Cy = -2x + 9
- Dy = 1/2 x - 4
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Answer: y = -2x + 9
The slope of k is (-7 - 5) / (4 - (-2)) = -12 / 6 = -2, and parallel lines share a slope. The choice with slope -1/2 is perpendicular to k rather than parallel, and the positive slopes reverse the sign.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 69Absolute valueChallenge
For how many integer values of x is |x - 4| < 3 true?
- A3
- B7
- C6
- D5
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Answer: 5
The inequality says x - 4 lies strictly between -3 and 3, so x lies strictly between 1 and 7. The integers in that range are 2, 3, 4, 5 and 6 — five of them. Counting 1 and 7 as well gives 7, but the inequality is strict.
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A recipe uses 350 g of flour for 12 scones. How much flour is needed for 30 scones?
- A840 g
- B1,050 g
- C875 g
- D292 g
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Answer: 875 g
Set up the proportion 350/12 = x/30, so x = 350 x 30 / 12 = 875 g. Tripling the 350 g gives 1,050, which would be right for 36 scones rather than 30.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 71Equivalent expressionsStandard
If 5(2x - 3) - 4(x - 2) = ax + b for every value of x, enter the value of a.
Student-produced response: write your own answer rather than choosing one.
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Answer: 6
Expanding gives 10x - 15 - 4x + 8 = 6x - 7, so a = 6 and b = -7. The commonest slip is distributing the -4 to only the first term inside its bracket, which leaves a = 6 but the wrong constant.
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A linear function f satisfies f(2) = 11 and f(7) = 26. What is f(0)?
- A8
- B3
- C11
- D5
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Answer: 5
The slope is (26 - 11) / (7 - 2) = 3. Working back from f(2) = 11 by two steps of 3 gives f(0) = 11 - 6 = 5. The choice 3 is the slope itself and 11 is the value at x = 2.
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What is the slope of the line 4x + 2y = 10?
- A-2
- B2
- C4
- D-1/2
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Answer: -2
Rearranged into y = mx + b: 2y = -4x + 10, so y = -2x + 5 and the slope is -2. Reading the coefficient of x straight off the original equation gives 4, which ignores the 2 attached to y.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 74Interpreting linear modelsStandard
A gym charges M(w) = 25 + 12w dollars for a membership lasting w weeks. What does the 12 represent?
- AThe cost of joining
- BThe cost of each additional week
- CThe number of weeks in a membership
- DThe largest a membership can cost
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Answer: The cost of each additional week
The 12 multiplies the number of weeks, so it is the amount added for each one. The 25 is the joining cost, paid whatever the length, and nothing in the model sets a maximum.
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Which inequality describes every number whose distance from 7 on the number line is at most 3?
- A|x + 7| <= 3
- B|x - 3| <= 7
- C|x - 7| <= 3
- D|x - 7| >= 3
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Answer: |x - 7| <= 3
Distance from 7 is written |x - 7|, and "at most 3" makes it less than or equal to 3. The plus sign would measure distance from -7, and the reversed inequality describes the numbers further than 3 away.
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The quantity y varies directly with x, and y = 45 when x = 12. What is y when x = 20?
- A53
- B27
- C108
- D75
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Answer: 75
The constant is 45 / 12 = 3.75, so y = 3.75 x 20 = 75. Adding 8 to y because x rose by 8 gives 53, which treats a proportional relationship as if it moved in step.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 77Function notationChallenge
If f(x) = 2x - 7 and g(x) = x^2, what is the value of f(g(3))?
- A11
- B1
- C4
- D-1
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Answer: 11
Work outward from the inside: g(3) = 9, then f(9) = 18 - 7 = 11. Doing it in the other order gives g(f(3)) = g(-1) = 1, which is why the order in the notation matters.
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For which value of c does the system 3x - y = 4 and 6x - 2y = c have infinitely many solutions?
- A4
- B8
- C2
- D0
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Answer: 8
Infinitely many solutions means the two equations describe the same line. The second is the first multiplied by 2 on the left, so the right must be doubled as well: c = 8. Any other value gives two parallel lines and no solution at all.
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Enter the greatest integer value of x for which 5x - 8 < 27.
Student-produced response: write your own answer rather than choosing one.
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Answer: 6
Adding 8 gives 5x < 35, so x < 7. The inequality is strict, so 7 itself does not satisfy it and the greatest integer that does is 6.
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A map is drawn to a scale of 1 cm to 25 km. Two towns are 14 cm apart on the map. Enter the distance between them in kilometres.
Student-produced response: write your own answer rather than choosing one.
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Answer: 350
Each centimetre stands for 25 km, so 14 cm stands for 14 x 25 = 350 km.
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What is the y-intercept of the line 3x - 5y = 20?
- A-4
- B4
- C20
- D-20
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Answer: -4
Set x = 0: -5y = 20, so y = -4. Rearranging into y = (3/5)x - 4 shows the same thing. Reading 20 off the equation forgets that it has not yet been divided by the coefficient of y.
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What is the solution to 2(x - 4) = 3x + 1?
- A-7
- B-9
- C9
- D7
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Answer: -9
Expanding gives 2x - 8 = 3x + 1. Subtracting 2x from both sides leaves -8 = x + 1, so x = -9. Check it: 2(-13) = -26 and 3(-9) + 1 = -26.
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If f(x) = |x - 6|, what is the value of f(2)?
- A-4
- B8
- C4
- D2
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Answer: 4
Substituting gives |2 - 6| = |-4| = 4. Absolute value returns a distance, so it is never negative, which rules out the first choice without any arithmetic.
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If 3x + 2y = 19 and x - y = 3, what is the value of y?
- A5
- B3
- C19
- D2
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Answer: 2
The second equation gives x = y + 3. Substituting: 3(y + 3) + 2y = 19, so 5y + 9 = 19 and y = 2. The other choices are x, the constant from the second equation, and the total from the first.
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A line passes through (0, -3) and (4, 5). What is its equation?
- Ay = 2x + 3
- By = -2x - 3
- Cy = 2x - 3
- Dy = (1/2)x - 3
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Answer: y = 2x - 3
The slope is (5 - (-3)) / (4 - 0) = 8 / 4 = 2, and the point (0, -3) is the y-intercept, so y = 2x - 3. Inverting the slope calculation gives the 1/2, and dropping the sign on the intercept gives y = 2x + 3.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 86Linear inequalitiesChallenge
Which describes every solution to -3x + 7 >= 22?
- Ax >= -5
- Bx >= 5
- Cx <= 5
- Dx <= -5
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Answer: x <= -5
Subtracting 7 gives -3x >= 15. Dividing by a negative reverses the inequality, so x <= -5. Test x = -6: -3(-6) + 7 = 25, which is at least 22. Forgetting to flip the sign is what this question is built around.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 87Equivalent expressionsStandard
If 4(2x + 3) - 5x = ax + b for every value of x, enter the value of b.
Student-produced response: write your own answer rather than choosing one.
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Answer: 12
Expanding gives 8x + 12 - 5x = 3x + 12, so a = 3 and b = 12. The constant comes only from the 4 times 3; the -5x contributes nothing to it.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 88Function notationStandard
The function g is defined by g(x) = 3x + 2. Enter the value of x for which g(x) = 20.
Student-produced response: write your own answer rather than choosing one.
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Answer: 6
Set 3x + 2 = 20, so 3x = 18 and x = 6. The question asks for the input, not the output, which is the step worth reading twice.
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What is the solution to 3(x + 2) = 4x - 5?
- A1
- B11
- C-11
- D3
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Answer: 11
Expanding gives 3x + 6 = 4x - 5. Subtracting 3x leaves 6 = x - 5, so x = 11. Check: 3(13) = 39 and 4(11) - 5 = 39.
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What is the slope of the line through (-1, 4) and (3, -4)?
- A-1/2
- B2
- C-2
- D1/2
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Answer: -2
The change in y is -4 - 4 = -8 and the change in x is 3 - (-1) = 4, so the slope is -8/4 = -2. Dividing the other way round gives -1/2, and dropping the minus signs gives the positive answers.
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What is the sum of all solutions to |3x + 6| = 15?
- A3
- B-7
- C10
- D-4
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Answer: -4
The expression inside is 15 or -15. From 3x + 6 = 15 comes x = 3; from 3x + 6 = -15 comes x = -7. Their sum is -4. The first two choices are the individual solutions, which is what stopping after one branch leaves you with.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 92Linear inequalitiesStandard
Which describes every solution to 4 - 2x < 10?
- Ax > -3
- Bx < -3
- Cx < 3
- Dx > 3
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Answer: x > -3
Subtracting 4 gives -2x < 6. Dividing by -2 reverses the inequality, so x > -3. Test x = -2: 4 + 4 = 8, which is less than 10. Failing to flip the sign gives x < -3, which fails that test.
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If f(x) = x/2 + 7, what is the value of f(12)?
- A19
- B13
- C9.5
- D26
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Answer: 13
Substituting gives 12/2 + 7 = 6 + 7 = 13. Adding before dividing gives 9.5, which is the order-of-operations slip this question tests.
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The quantity y varies directly with x, and y = 84 when x = 7. What is y when x = 11?
- A88
- B96
- C132
- D122
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Answer: 132
The constant of variation is 84 / 7 = 12, so y = 12 x 11 = 132. Adding 4 to y because x rose by 4 gives 88, which would be right only if the relationship were "y = x + something".
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If 2x + y = 13 and y = x + 1, enter the value of x.
Student-produced response: write your own answer rather than choosing one.
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Answer: 4
Substituting the second equation into the first gives 2x + x + 1 = 13, so 3x = 12 and x = 4 (with y = 5). Check: 8 + 5 = 13.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 96Interpreting linear modelsStandard
A tank contains W(d) = 900 - 24d litres of water d days after it was last filled. Enter the number of litres lost each day.
Student-produced response: write your own answer rather than choosing one.
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Answer: 24
The coefficient of d is the change for each additional day, and the minus sign shows it is a loss, so 24 litres a day. The 900 is the starting volume.
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If 2x/3 = 8, what is the value of x?
- A16
- B5.33
- C24
- D12
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Answer: 12
Multiplying both sides by 3 gives 2x = 24, so x = 12. Multiplying by 3 without then halving gives 24, which is the step most often left out.
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A line has a y-intercept of 5 and a slope of -3. What is y when x = 4?
- A-7
- B17
- C2
- D-12
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Answer: -7
Substituting into y = -3x + 5 gives -12 + 5 = -7. Dropping the sign on the slope gives 17, and forgetting the intercept gives -12.
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Which describes every solution to |x + 2| >= 6?
- Ax >= 4
- Bx <= -8 or x >= 4
- C-8 <= x <= 4
- Dx <= -8
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Answer: x <= -8 or x >= 4
The expression inside is at least 6 away from zero in either direction, so x + 2 >= 6 or x + 2 <= -6, giving x >= 4 or x <= -8. A "greater than" absolute value splits into two rays; only "less than" gives a single interval.
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What is the x-intercept of the line 5x - 2y = 20?
- A4
- B-10
- C10
- D-4
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Answer: 4
The x-intercept is where y = 0, so 5x = 20 and x = 4. Setting x to zero instead gives the y-intercept, -10, which is the commonest confusion here.
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If f(x) = 4 - x^2, what is the value of f(-3)?
- A13
- B-5
- C5
- D1
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Answer: -5
Substituting gives 4 - (-3)^2 = 4 - 9 = -5. The square makes the input positive before it is subtracted; treating the minus as part of the square gives 13.
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If 4x + y = 17 and y = 2x - 1, what is the value of x?
- A4
- B2
- C3
- D6
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Answer: 3
Substituting the second into the first gives 4x + 2x - 1 = 17, so 6x = 18 and x = 3 (with y = 5). Check: 12 + 5 = 17.
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Forty percent of a number is 26. Enter the number.
Student-produced response: write your own answer rather than choosing one.
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Answer: 65
If 0.40n = 26 then n = 26 / 0.40 = 65. Taking 40 percent of 26 answers the reverse question and gives 10.4.
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If 3(2x - 1) - 2(x - 4) = ax + b for every value of x, enter the value of a.
Student-produced response: write your own answer rather than choosing one.
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Answer: 4
Expanding gives 6x - 3 - 2x + 8 = 4x + 5, so a = 4 and b = 5. The sign on the second bracket matters twice: -2 times -4 adds 8.
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What is the solution to 7 - 2x = x + 1?
- A-2
- B8/3
- C6
- D2
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Answer: 2
Add 2x to both sides: 7 = 3x + 1. Subtract 1: 6 = 3x, so x = 2. Check: 7 - 4 = 3 and 2 + 1 = 3.
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What is the slope of the line through (0, -2) and (5, 13)?
- A3
- B1/3
- C-3
- D15
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Answer: 3
The change in y is 13 - (-2) = 15 and the change in x is 5, so the slope is 15/5 = 3. The choice 15 stops before dividing.
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What is the product of all solutions to |x - 9| = 4?
- A18
- B65
- C36
- D13
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Answer: 65
The expression inside is 4 or -4, giving x = 13 and x = 5. Their product is 65. The choices 13 and 18 are one solution and the sum, both of which come from stopping early.
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What is the y-intercept of the line 3x + 2y = 12?
- A4
- B12
- C6
- D-6
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Answer: 6
The y-intercept is where x = 0, so 2y = 12 and y = 6. Setting y to zero instead gives the x-intercept of 4, which is the usual confusion.
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If f(x) = 6 - 2x, what is the value of f(-4)?
- A-2
- B2
- C-14
- D14
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Answer: 14
Substituting gives 6 - 2(-4) = 6 + 8 = 14. Subtracting a negative adds, which is the whole of this question.
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If 2x + 3y = 21 and y = x + 2, what is the value of y?
- A5
- B3
- C7
- D9
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Answer: 5
Substituting gives 2x + 3(x + 2) = 21, so 5x + 6 = 21 and x = 3, making y = 5. Check: 6 + 15 = 21. The choice 3 is x rather than y.
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Enter the solution to 5(x - 2) = 3x + 4.
Student-produced response: write your own answer rather than choosing one.
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Answer: 7
Expanding gives 5x - 10 = 3x + 4. Subtracting 3x leaves 2x - 10 = 4, so 2x = 14 and x = 7.
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The quantity y varies directly with x, and y = 54 when x = 9. Enter the value of y when x = 15.
Student-produced response: write your own answer rather than choosing one.
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Answer: 90
The constant is 54 / 9 = 6, so y = 6 x 15 = 90. Adding 6 to y because x rose by 6 would give 60, which treats a proportional relationship as an additive one.
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