Maths: Algebra & Equations

Solve 3x + 5 = 29.

• Ax = 9
• Bx = 8✓
• Cx = 6
• Dx = 11

Answer: Subtract 5 from both sides to get 3x = 24, then divide by 3: x = 8. Checking: 3 × 8 + 5 = 29.

Expand and simplify 3(2x − 4) − 2(x + 5).

• A4x − 22✓
• B4x − 2
• C4x + 2
• D8x − 22

Answer: 3(2x − 4) = 6x − 12 and −2(x + 5) = −2x − 10. Adding them: 6x − 2x = 4x and −12 − 10 = −22, giving 4x − 22. Sign errors on the second bracket produce the other answers.

Factorise fully: 6x² + 9x.

• A3x(2x + 3)✓
• B3(2x² + 3x)
• Cx(6x + 9)
• D3x(2x + 9)

Answer: The highest common factor of 6x² and 9x is 3x, so 6x² + 9x = 3x(2x + 3). The first two alternatives take out only part of the common factor, and the last one expands to 6x² + 27x.

If a = 4 and b = −2, what is the value of 5a − 3b?

• A−26
• B26✓
• C22
• D14

Answer: 5a = 20 and 3b = −6, so 5a − 3b = 20 − (−6) = 26. Subtracting a negative adds; treating 3b as +6 gives 14.

Solve the simultaneous equations 2x + y = 11 and x − y = 1.

• Ax = 3, y = 5
• Bx = 4, y = 5
• Cx = 4, y = 3✓
• Dx = 5, y = 1

Answer: Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting into x − y = 1 gives y = 3. Check the first equation: 2 × 4 + 3 = 11. Each alternative satisfies only one of the two equations.

The nth term of a sequence is 4n − 1. Which of these is the 12th term?

• A43
• B47✓
• C48
• D44

Answer: Substitute n = 12: 4 × 12 − 1 = 48 − 1 = 47. 43 is the 11th term, 48 forgets the −1, and 44 is 4n − 4.

Simplify 5a + 3b − 2a + 7b.

• A3a + 10b✓
• B7a + 10b
• C3a + 4b
• D13ab

Answer: Collect like terms: 5a − 2a = 3a and 3b + 7b = 10b. Terms in a and terms in b cannot be combined with each other, so 13ab is meaningless here.

Solve 4(x − 2) = 2x + 6.

• Ax = 7✓
• Bx = −7
• Cx = 1
• Dx = 4

Answer: Expand: 4x − 8 = 2x + 6. Subtract 2x from both sides: 2x − 8 = 6, so 2x = 14 and x = 7. Check: 4 × 5 = 20 and 2 × 7 + 6 = 20.

Make x the subject of the formula y = 3x + 5.

• Ax = 3y − 5
• Bx = y ÷ 3 − 5
• Cx = (y + 5) ÷ 3
• Dx = (y − 5) ÷ 3✓

Answer: Subtract 5 from both sides (y − 5 = 3x), then divide by 3: x = (y − 5) ÷ 3. The bracket matters: dividing only y by 3 before subtracting gives a different, wrong formula.

Factorise x² + 5x + 6.

• A(x + 1)(x + 6)
• B(x + 2)(x + 3)✓
• C(x − 2)(x − 3)
• D(x + 5)(x + 1)

Answer: Look for two numbers that multiply to 6 and add to 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). The pair 1 and 6 multiplies to 6 but adds to 7, and (x − 2)(x − 3) expands to x² − 5x + 6.

Find the value of 2x² − 3x when x = 3.

• A3
• B27
• C9✓
• D18

Answer: Square first, then multiply: 2 × 3² = 2 × 9 = 18, and 3 × 3 = 9, so 18 − 9 = 9. Squaring 2x first (6² = 36) instead of x is the common error.

Which inequality is shown by the solutions x = −1, 0, 1, 2, 3 (integers only)?

• A−2 < x ≤ 3✓
• B−1 ≤ x ≤ 2
• C−2 ≤ x < 3
• D−1 < x < 3

Answer: The integers from −1 to 3 inclusive satisfy −2 < x ≤ 3: −2 is excluded by the strict inequality and 3 is included. The other inequalities exclude −1, exclude 3, or stop at 2.