consider the unfinished equation 12(x - 3) + 18 = match the expressions with number of solutions the equation would have with that expression on the right hand

6(2x-3)
4(3x-3)
4(2x-3)

1 answer

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.

  1. 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).

  2. 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.

  3. 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.