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 = - 3 and b = - 5 then there are infinitely many solutions to the equation.
If a = 9 and then there is exactly one solution to the equation. b = 7
If and then there is no solution to the equation. a = - 9 b = 13
If a = - 3 and b = 4 then there is exactly one solution to the equation.
If a = 9 and then there is no solution to the equation. b = - 2

Answer 1

Simplify the left side: 3(3x+4)-5 = 9x+7. So the equation is 9x+7 = ax + b.

General rule:
- If a ≠ 9: one solution (x = (b−7)/(9−a)).
- If a = 9 and b = 7: infinitely many solutions (identity).
- If a = 9 and b ≠ 7: no solution (contradiction).

Evaluate each case:
- a = −3, b = −5: a ≠ 9 ⇒ exactly one solution. (Statement saying infinitely many is false.)
- a = 9, b = 7: a = 9 and b = 7 ⇒ infinitely many solutions (so the statement claiming exactly one is false).
- a = −9, b = 13: a ≠ 9 ⇒ exactly one solution. (Statement saying no solution is false.)
- a = −3, b = 4: a ≠ 9 ⇒ exactly one solution. (True.)
- a = 9, b = −2: a = 9 and b ≠ 7 ⇒ no solution. (True.)

True statements: the ones about (a = −3, b = 4) and (a = 9, b = −2).