Question

Consider the equation 4x - 5(2x + 1) = ax + b
Part A
Determine whether each pair of values for a and b results in the equation having no solution or one solution.
Select No Solution or One Solution for each pair of values.
a = - 6 b = 0 No Solution One Solution
b = - 5 a = 0 No Solution One Solution
a = 1 b = - 5 No Solution One Solution
a = - 6 b = 5 No Solution One Solution

Answer 1

First simplify: 4x − 5(2x + 1) = 4x − 10x − 5 = −6x − 5, so the equation is
−6x − 5 = ax + b ⇔ (−6 − a)x + (−5 − b) = 0.

- If −6 − a ≠ 0 (i.e. a ≠ −6) there is one solution.
- If −6 − a = 0 (a = −6) and −5 − b ≠ 0 (b ≠ −5) there is no solution.
(If a = −6 and b = −5 there would be infinitely many solutions.)

Now the pairs:

- a = −6, b = 0 → No Solution
- a = 0, b = −5 → One Solution
- a = 1, b = −5 → One Solution
- a = −6, b = 5 → No Solution