Question

Which equation is equivalent to 3 + 5 (x + 2) = 12
1. 1 + 3 (x+2) = 1
2. 5 (x+2) -x = 1
3. 18 + 6 (x-2) - 2x = 1
4. 9x - 5(x-1) - 1 = 1

Answer 1

To determine which equation is equivalent to \( 3 + 5 (x + 2) = 12 \), we can simplify the original equation.

Starting with the original equation:

1. Distribute \( 5 \) across \( (x + 2) \):
\[
3 + 5x + 10 = 12
\]
2. Combine \( 3 \) and \( 10 \):
\[
5x + 13 = 12
\]
3. Subtract \( 13 \) from both sides:
\[
5x = 12 - 13
\]
\[
5x = -1
\]
4. Divide by \( 5 \):
\[
x = -\frac{1}{5}
\]

Now we can evaluate each of the provided equations to see if they simplify to \( x = -\frac{1}{5} \).

**Option 1: \( 1 + 3(x + 2) = 1 \)**

- Simplifying leads to:
\[
3(x + 2) = 0
\]
\[
x + 2 = 0 \implies x = -2 \quad \text{(not equivalent)}
\]

**Option 2: \( 5(x + 2) - x = 1 \)**

- Simplifying leads to:
\[
5x + 10 - x = 1
\]
\[
4x + 10 = 1 \implies 4x = 1 - 10 \implies 4x = -9 \implies x = -\frac{9}{4} \quad \text{(not equivalent)}
\]

**Option 3: \( 18 + 6(x - 2) - 2x = 1 \)**

- Simplifying leads to:
\[
18 + 6x - 12 - 2x = 1
\]
\[
4x + 6 = 1 \implies 4x = 1 - 6 \implies 4x = -5 \implies x = -\frac{5}{4} \quad \text{(not equivalent)}
\]

**Option 4: \( 9x - 5(x - 1) - 1 = 1 \)**

- Simplifying leads to:
\[
9x - 5x + 5 - 1 = 1
\]
\[
4x + 4 = 1 \implies 4x = 1 - 4 \implies 4x = -3 \implies x = -\frac{3}{4} \quad \text{(not equivalent)}
\]

None of the options simplify to the same solution \( x = -\frac{1}{5} \).

Since our evaluation may have overlooked the second equation's simplification, let’s check it again in detail:

Re-evaluating **Option 2** :
1. \( 5(x + 2) - x = 1 \)
\[
5x + 10 - x = 1
\]
\[
4x + 10 = 1
\]
\[
4x = 1 - 10
\]
\[
4x = -9
\]
\[
x = -\frac{9}{4}
\]

This confirms that the options might still be leading to incorrect evaluations.

Since computed values haven’t matched, let’s join different positions by confirming the five options aren’t reaching the starting equation equivalency. Each proposed equation diverges from the original finding.

Thus, the answer is:

**No options presented are equivalent to \( 3 + 5 (x + 2) = 12 \).**