Question

Which of the following equations is equivalent to 4x + 6 = 1?

A. 1 + 3(x + 2) = 1

B. 5(x + 2) - x = 1

C. 18 + 6(x - 2) - 2x = 1

D. 9x - 5(x - 1) - 1 = 1

Answer 1

To determine which of the provided equations is equivalent to \(4x + 6 = 1\), we can start by rearranging the original equation to form a standard equation for comparison.

1. Start with the original equation:

\[
4x + 6 = 1
\]

2. Subtract 6 from both sides:

\[
4x = 1 - 6
\]
\[
4x = -5
\]

3. Now, we can solve for \(x\) by dividing both sides by 4:

\[
x = -\frac{5}{4}
\]

Next, we'll evaluate each option to see if they also yield \(x = -\frac{5}{4}\).

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

Set the equation:

\[
1 + 3(x + 2) = 1
\]

Subtract 1 from both sides:

\[
3(x + 2) = 0
\]

Divide by 3:

\[
x + 2 = 0
\]

Subtract 2:

\[
x = -2
\]
(This does not match \(-\frac{5}{4}\).)

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

Set the equation:

\[
5(x + 2) - x = 1
\]

Distribute the 5:

\[
5x + 10 - x = 1
\]

Combine like terms:

\[
4x + 10 = 1
\]

Subtract 10 from both sides:

\[
4x = 1 - 10
\]
\[
4x = -9
\]

Divide by 4:

\[
x = -\frac{9}{4}
\]
(This does not match \(-\frac{5}{4}\).)

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

Set the equation:

\[
18 + 6(x - 2) - 2x = 1
\]

Distribute the 6:

\[
18 + 6x - 12 - 2x = 1
\]

Combine like terms:

\[
(6x - 2x) + (18 - 12) = 1
\]
\[
4x + 6 = 1
\]

This is the same as the original equation! (This matches our rearranged equation.)

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

Set the equation:

\[
9x - 5(x - 1) - 1 = 1
\]

Distribute the -5:

\[
9x - 5x + 5 - 1 = 1
\]

Combine like terms:

\[
(9x - 5x) + (5 - 1) = 1
\]
\[
4x + 4 = 1
\]

Subtract 4 from both sides:

\[
4x = 1 - 4
\]
\[
4x = -3
\]

Divide by 4:

\[
x = -\frac{3}{4}
\]
(This does not match \(-\frac{5}{4}\).)

**Conclusion:** The only option that is equivalent to the original equation \(4x + 6 = 1\) is:

**C. \(18 + 6(x - 2) - 2x = 1\)**.