To determine the number of solutions for the unfinished equation \(12(x - 3) + 18 = \) (expression), we first simplify the left-hand side.
Starting with the left side:
\[ 12(x - 3) + 18 \] Distributing \(12\):
\[ = 12x - 36 + 18 \] \[ = 12x - 18 \]
Now, we need to match the expressions \(6(2x - 3)\), \(4(3x - 3)\), and \(4(2x - 3)\) with the simplified left-hand side \(12x - 18\) in order to determine the number of solutions for each equation formed.
-
For the expression \(6(2x - 3)\):
Simplifying \(6(2x - 3)\):
\[ = 12x - 18 \] So the equation \(12x - 18 = 12x - 18\) has infinitely many solutions (since both sides are identical).
-
For the expression \(4(3x - 3)\):
Simplifying \(4(3x - 3)\):
\[ = 12x - 12 \] So the equation \(12x - 18 = 12x - 12\) simplifies to:
\[ -18 \neq -12 \] Thus, this equation has no solutions.
-
For the expression \(4(2x - 3)\):
Simplifying \(4(2x - 3)\):
\[ = 8x - 12 \] So the equation \(12x - 18 = 8x - 12\) simplifies to:
\[ 12x - 8x = -12 + 18 \] \[ 4x = 6 \] \[ x = \frac{3}{2} \] This equation has exactly one solution.
Summary of Solutions:
- \(6(2x - 3)\): Infinitely many solutions.
- \(4(3x - 3)\): No solutions.
- \(4(2x - 3)\): Exactly one solution.
1 answer