Question

Consider solving the equation for 2.
3 (3x + 4) - 5 = ax + b
Which statements are true about the solution to the equation when substituting values for a and b as specified?
Select all that apply.
A: a = - 3 and b = - 5 then there are infinitely many solutions to the equation.
B: If a = 9 and then there is exactly one solution to the equation. b = 7
C:If and then there is no solution to the equation. a = - 9 b = 13
D: If a = - 3 and b = 4 then there is exactly one solution to the equation.
E: If a = 9 and then there is no solution to the equation. b = - 2

Answer 1

First simplify the equation:
3(3x + 4) - 5 = 9x + 12 - 5 = 9x + 7,
so the equation is 9x + 7 = ax + b.

- If a = 9 and b = 7, the two sides are identical for all x → infinitely many solutions.
- If a = 9 and b ≠ 7, the equation reduces to a contradiction (e.g. 7 = b with b ≠ 7) → no solution.
- If a ≠ 9, there is exactly one solution x = (b − 7)/(9 − a).

Check each choice:
A: a = −3, b = −5 → a ≠ 9 so one solution (not infinitely many). False.
B: a = 9, b = 7 → gives infinitely many, not exactly one. False.
C: a = −9, b = 13 → a ≠ 9 so one solution, not none. False.
D: a = −3, b = 4 → a ≠ 9 so exactly one solution (x = (4−7)/(9−(−3)) = −1/4). True.
E: a = 9, b = −2 → a = 9 but b ≠ 7 gives no solution. True.

Correct answers: D and E.