Solve 4x - 13 = 17 + 8x
x =
Solving linear equations — Notes & Worked Examples Show / Hide
Explanation:
To solve a linear equation, use inverse operations on both sides of the equals sign to isolate the unknown letter x. If the equation contains brackets, expand them first. If there is a fraction, multiply both sides by the denominator to clear it.
When the unknown x appears on both sides of the equation, start by eliminating the smaller x term from both sides so that you keep a positive number of x's, then solve the remaining two-step equation.
Example 1: Solve 7x − 4 = 3x + 18
Solution:
Subtract 3x from both sides: 4x − 4 = 18
Add 4 to both sides: 4x = 22
Divide both sides by 4: x = 224 = 5.5
Answer: x = 5.5
Example 2: Solve 3(2x − 1)5 = 9
Solution:
Multiply both sides by 5: 3(2x − 1) = 45
Divide both sides by 3: 2x − 1 = 15
Add 1 to both sides: 2x = 16
Divide by 2: Answer: x = 8
Solve 5x + 11 = 3x + 19
x =
Worked Solution & Answer
Solve 5x - 6 = 3(x - 1)
x =
Worked Solution & Answer
5x – 6 = 3x – 3 M1
⇒ 2x = 3 M1
⇒ x = 1.5 (or 1 1/2) A1
⇒ 2x = 3 M1
⇒ x = 1.5 (or 1 1/2) A1
Answer: x = 1.5
Solve 2(4x - 5) = 18
x =
Worked Solution & Answer
8x – 10 = 18 M1
⇒ 8x = 28 M1
⇒ x = 28 ÷ 8 = 3.5 A1
⇒ 8x = 28 M1
⇒ x = 28 ÷ 8 = 3.5 A1
Answer: x = 3.5
Solve 4(2x - 3) = 20
x =
Worked Solution & Answer
Solve 4(3x - 2) = 2x - 5
x =
Worked Solution & Answer
12x - 8 = 2x - 5 M1
12x - 2x = -5 + 8 M1
10x = 3
x = 0.3 (or 310) A1
12x - 2x = -5 + 8 M1
10x = 3
x = 0.3 (or 310) A1
Answer: x = 0.3
Solve 3(2 - 4x) = 5 - 8x
Show clear algebraic working.
Show clear algebraic working.
x =
Worked Solution & Answer
6 − 12x = 5 − 8x M1
⇒ 6 − 5 = 12x − 8x M1
⇒ 4x = 1
⇒ x = 1/4 (or 0.25). A1
⇒ 6 − 5 = 12x − 8x M1
⇒ 4x = 1
⇒ x = 1/4 (or 0.25). A1
Answer: x = 14
(a) Solve m – 3 = 4 (1) (b) Solve 3n + n = 24 (2)
(a) Solve m - 3 = 4 [1 mark]
m =
(b) Solve 3n + n = 24 [2 marks]
n =
Worked Solution & Answer
(a)
m = 4 + 3 = 7 B1
Answer (a): m = 7
(b)
4n = 24 M1
⇒ n = 6 A1
⇒ n = 6 A1
Answer (b): n = 6
(a) Solve 4(x – 5) = 18 (2) (b) –3 < t ⩽ 2, t is an integer. Write down all possible values of t. (2)
(a) Solve 4(x - 5) = 18 [2 marks]
x =
(b) Write down all the possible values of t. [2 marks]
Worked Solution & Answer
(a)
4x – 20 = 18 M1
⇒ 4x = 38
⇒ x = 9.5 A1
⇒ 4x = 38
⇒ x = 9.5 A1
Answer (a): x = 9.5
(b)
Integers strictly greater than –3 and up to 2: –2, –1, 0, 1, 2 M1 A1
Answer (b): –2, –1, 0, 1, 2
Solve the following equations.
(a) Solve 11x - 3 = 6x + 1 [3 marks]
x =
(b) Solve 2x5 = 14 [2 marks]
x =
Worked Solution & Answer
(a)
11x - 6x = 1 + 3 ⇒ 5x = 4 ⇒ x = 45 = 0.8 M2 A1
Answer (a): x = 0.8
(b)
2x = 14 × 5 = 70 ⇒ x = 702 = 35 M1 A1
Answer (b): x = 35