Equations (Copy)

Practice Questions — 2.5 Equations

Question 1

Solve for x:
(a) 3x + 4 = 10
(b) 5 − 2x = 3(x + 7)
(c) 7x − 12 = 2x + 8

Question 2

Construct and simplify an expression for:
(a) The product of two consecutive odd numbers.
(b) The sum of two consecutive even numbers.
(c) The difference between the squares of two consecutive integers.

Question 3

Solve for x:
(a) (x)/(2x + 1) = 4
(b) (2/(x + 2)) + (3/(2x − 1)) = 1
(c) x/(x + 2) = 3/(x − 6)

Question 4

Solve simultaneously for x and y:
(a) 2x + 3y = 12 and x − y = 4
(b) 3x + 2y = 16 and 2x − y = 1
(c) 4x − y = 11 and 2x + 3y = 19

Question 5

Solve by factorisation:
(a) x² + 5x + 6 = 0
(b) y² − 7y + 12 = 0
(c) p² − 9p + 20 = 0

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Question 6

Solve by completing the square:
(a) x² + 6x + 5 = 0
(b) y² − 4y − 5 = 0
(c) m² + 8m + 11 = 0

Question 7

Solve using the quadratic formula:
(a) x² − 2x − 8 = 0
(b) 2y² + 3y − 2 = 0
(c) 3p² − 5p + 2 = 0

Question 8

Change the subject of the formula:
(a) A = 2x + 3y, make x the subject.
(b) V = (1/3)πr²h, make h the subject.
(c) F = ma, make m the subject.

Question 9

Change the subject of the formula where the subject appears more than once:
(a) y = x + 2/x, make x the subject.
(b) s = (2x − 3)/(x + 4), make x the subject.
(c) P = (2ab)/(a + b), make a the subject.

Question 10

Form and solve equations from word problems:
(a) The sum of two consecutive integers is 65. Find the integers.
(b) Twice a number increased by 7 equals 29. Find the number.
(c) The area of a rectangle is 48 cm². If its length is 2 more than its breadth, find the dimensions.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Answer key and explanations — 2.5 Equations

1. Solve for x

(a) 3x + 4 = 10 → 3x = 6 → x = 2
(b) 5 − 2x = 3(x + 7) → 5 − 2x = 3x + 21 → −5x = 16 → x = −16/5
(c) 7x − 12 = 2x + 8 → 5x = 20 → x = 4
Explanation: Collect like terms and isolate x.

2. Construct expressions

(a) Two consecutive odd numbers: n and n + 2 → product = n(n + 2) = n² + 2n
(b) Two consecutive even numbers: 2k and 2k + 2 → sum = 4k + 2
(c) Two consecutive integers: m and m + 1 → difference of squares = (m + 1)² − m² = 2m + 1

3. Fractional equations

(a) x/(2x + 1) = 4 → x = 8x + 4 → −7x = 4 → x = −4/7
(b) 2/(x + 2) + 3/(2x − 1) = 1
→ Multiply through: [2(2x − 1) + 3(x + 2)] / [(x + 2)(2x − 1)] = 1
→ (4x − 2 + 3x + 6) = (x + 2)(2x − 1)
→ 7x + 4 = 2x² + 3x − 2 → 2x² − 4x − 6 = 0 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0
→ x = 3 or −1 (check denominators: both valid).
(c) x/(x + 2) = 3/(x − 6)
→ cross multiply: x(x − 6) = 3(x + 2)
→ x² − 6x = 3x + 6
→ x² − 9x − 6 = 0
Solve: x = [9 ± √(81 + 24)]/2 = [9 ± √105]/2
Solutions: (9 ± √105)/2

4. Simultaneous equations

(a) 2x + 3y = 12, x − y = 4
From second: x = y + 4. Sub into first: 2(y + 4) + 3y = 12 → 2y + 8 + 3y = 12 → 5y = 4 → y = 4/5, x = 24/5.
(b) 3x + 2y = 16, 2x − y = 1
From second: y = 2x − 1. Sub: 3x + 2(2x − 1) = 16 → 3x + 4x − 2 = 16 → 7x = 18 → x = 18/7, y = 29/7.
(c) 4x − y = 11, 2x + 3y = 19
From first: y = 4x − 11. Sub: 2x + 3(4x − 11) = 19 → 2x + 12x − 33 = 19 → 14x = 52 → x = 26/7, y = 15/7.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

5. Factorisation method

(a) x² + 5x + 6 = (x + 2)(x + 3) → x = −2, −3
(b) y² − 7y + 12 = (y − 3)(y − 4) → y = 3, 4
(c) p² − 9p + 20 = (p − 4)(p − 5) → p = 4, 5

6. Completing the square

(a) x² + 6x + 5 = 0 → (x + 3)² − 9 + 5 = 0 → (x + 3)² = 4 → x = −3 ± 2 → −1, −5
(b) y² − 4y − 5 = 0 → (y − 2)² − 4 − 5 = 0 → (y − 2)² = 9 → y = 2 ± 3 → 5, −1
(c) m² + 8m + 11 = 0 → (m + 4)² − 16 + 11 = 0 → (m + 4)² = 5 → m = −4 ± √5

7. Quadratic formula

(a) x² − 2x − 8 = 0
x = [2 ± √(4 + 32)]/2 = [2 ± √36]/2 = [2 ± 6]/2 → 4 or −2
(b) 2y² + 3y − 2 = 0
y = [−3 ± √(9 + 16)]/4 = [−3 ± 5]/4 → y = (2/4) = ½ or y = (−8/4) = −2
(c) 3p² − 5p + 2 = 0
p = [5 ± √(25 − 24)]/6 = [5 ± 1]/6 → 1 or 2/3

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

8. Change subject

(a) A = 2x + 3y → 2x = A − 3y → x = (A − 3y)/2
(b) V = (1/3)πr²h → h = 3V/(πr²)
(c) F = ma → m = F/a

9. Change subject (harder)

(a) y = x + 2/x
→ Multiply: yx = x² + 2 → x² − yx + 2 = 0
→ Solve quadratic in x: x = [y ± √(y² − 8)]/2
(b) s = (2x − 3)/(x + 4)
→ s(x + 4) = 2x − 3 → sx + 4s = 2x − 3 → (s − 2)x = −4s − 3 → x = (−4s − 3)/(s − 2)
(c) P = 2ab/(a + b)
→ P(a + b) = 2ab → Pa + Pb = 2ab → Pa − 2ab = −Pb → a(P − 2b) = −Pb → a = −Pb/(P − 2b)

10. Word problems

(a) n + (n + 1) = 65 → 2n + 1 = 65 → 2n = 64 → n = 32 → integers: 32 and 33
(b) 2x + 7 = 29 → 2x = 22 → x = 11
(c) Let breadth = x, length = x + 2 → area = x(x + 2) = 48 → x² + 2x − 48 = 0 → (x + 8)(x − 6) = 0 → x = 6 (positive solution) → breadth = 6, length = 8.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course