Question

Consider the unfinished equation 12(x - 3) + 18 = _______. Match the following expressions with the number of solutions the equation would have with that expression on the right-hand side.

(3 points)

6(2x-3)

4(3x-3)

4(2x-3)

answers are no solution, one solution, and infinite solutions

Answer 1

Let's analyze the given unfinished equation:

\[ 12(x - 3) + 18 = \_\_\_\_ \]

First, simplify the left-hand side:

\[ 12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18 \]

Now, we can write the complete equation as:

\[ 12x - 18 = y \]

where \( y \) is the expression on the right-hand side from the options provided.

Next, we will match each expression with the number of solutions based on how they can be simplified:

1. **For \( 6(2x-3) \)**:
\[
6(2x - 3) = 12x - 18
\]
Here, both sides are equal, so there are **infinite solutions**.

2. **For \( 4(3x-3) \)**:
\[
4(3x - 3) = 12x - 12
\]
This simplifies to:
\[
12x - 18 = 12x - 12 \implies -18 = -12
\]
This is a contradiction, which means there are **no solutions**.

3. **For \( 4(2x-3) \)**:
\[
4(2x - 3) = 8x - 12
\]
This simplifies to:
\[
12x - 18 = 8x - 12
\]
Rearranging gives:
\[
12x - 8x = -12 + 18 \implies 4x = 6 \implies x = \frac{3}{2}
\]
This gives us one solution.

So, the matches are:

- \( 6(2x-3) \): **infinite solutions**
- \( 4(3x-3) \): **no solution**
- \( 4(2x-3) \): **one solution**