Find the rational number x such that: 56(x + 35) = 56x + 12.
Answer: Opening the bracket gives 56x + 12 = 56x + 12, which is true for every value of x. So there is no single answer: every rational number x satisfies the equation (for example x = 0).
Step-by-step solution
Idea: Use the distributive property on the left side and compare with the right side. If the two sides become exactly the same, the equation holds for all x (it is an identity), not for one particular x.
- Distributive property: 56(x + 35) = 56x + 56 × 35.½ mark
- 56 × 35 = 1530 = 12. So the left side is 56x + 12.½ mark
- The equation becomes 56x + 12 = 56x + 12. Taking 56x + 12 from both sides leaves 0 = 0, which is always true.½ mark
- So the equation is true for every rational number x; there is no single special value. (As printed, the question has no unique answer. It shows the distributive property: the two sides are always equal.)½ mark
The two sides are identical after using the distributive property, so every rational number x satisfies the equation (e.g. x = 0, x = 1, x = −2/5 all work).
Check: x = 0: LHS = 56 × 35 = 12, RHS = 12 ✓. x = 1: LHS = 56 × 85 = 43, RHS = 56 + 12 = 43 ✓.
Answer to write in the exam
LHS = 56x + 56 × 35 = 56x + 12
56x + 12 = 56x + 12
0 = 0, true for all x
∴ Every rational number x satisfies the equation (e.g. x = 0).
Common mistakes that cost marks
- Multiplying 56 by x only and forgetting 56 × 35, which leads to a wrong ‘solution’.
- Reaching 0 = 0 and writing ‘x = 0′. 0 = 0 means the equation is true for all x, not that x is 0.
- Calculating 56 × 35 as 811 by adding.
How this can come in the exam
MCQ (1 mark)
The value of x for which 23(x + 14) = 23x + 12 is
- 0
- 14
- 34
- no value of x
Show answer
(D) no value of x
LHS = 23x + 16. Then 16 = 12 would be needed, which is false, so no x works.
Short answer (2 marks)
Solve 34(x + 23) = 12x + 1.
Show answer
34x + 12 = 12x + 1 (1 mark). 14x = 12, so x = 2 (1 mark).Try one yourself
Find x if 25(x + 52) = 15x + 2.
Show answer
25x + 1 = 15x + 2 ⇒ 15x = 1 ⇒ x = 5.
More questions like this
- Try and represent 85 and −74 on a number line.
- The absolute value of a rational number x, written as |x|, represents its distance from 0 on the number line. |53| = 53, |−53| is also equal to 53 and |0| = 0.
- Try to explain why the average of two rational numbers a and b, which equals (a + b)2, is always a rational number between a and b.
- This means that there are infinitely many rational numbers between any two points. It feels as though the rational numbers must completely fill the number line, leaving no gaps whatsoever. But do they?
- Represent the rational numbers 23, −54 and 112 on a single number line.