Question

Samuel is solving the equation 3 (x minus 1) + 4 = 2 minus (x + 3). Which equivalent equations might Samuel use? Check all that apply.
3 x minus 3 + 4 = 2 minus x minus 3
3 x minus 1 + 4 = 2 minus x + 3
3 x minus 3 + 4 = 2 minus x + 3
3 x + 1 = negative x minus 1
3 x + 3 = negative x + 5
3 x + 1 = negative x + 5

Answer 1

To determine which equations are equivalent to the original equation \( 3(x - 1) + 4 = 2 - (x + 3) \), we can simplify the original equation step-by-step:

1. Start with the original equation:
\[
3(x - 1) + 4 = 2 - (x + 3)
\]

2. Distribute \(3\) on the left side:
\[
3x - 3 + 4 = 2 - (x + 3)
\]

3. Combine like terms on the left side:
\[
3x + 1 = 2 - (x + 3)
\]

4. Simplify the right side:
\[
3x + 1 = 2 - x - 3
\]
\[
3x + 1 = -x - 1
\]

Now, we have derived two equivalent forms:
1. \( 3x + 1 = -x - 1 \)
2. We can also express \(2 - (x + 3)\) as \( -x - 1\), which confirms our formulation.

5. Further simplifying \( 2 - (x + 3) \):
\[
3x + 1 = -x - 1
\]

So, let's review the options:

1. **Option**: \( 3x - 3 + 4 = 2 - x - 3 \)
- Simplifying gives \( 3x + 1 = -x - 1 \). **This is equivalent.**

2. **Option**: \( 3x - 1 + 4 = 2 - x + 3 \)
- Simplifying gives \( 3x + 3 = -x + 5\). **This is NOT equivalent.**

3. **Option**: \( 3x - 3 + 4 = 2 - x + 3 \)
- This simplifies to \( 3x + 1 = -x + 5\). **This is NOT equivalent.**

4. **Option**: \( 3x + 1 = -x - 1 \)
- This is what we found in simplification. **This is equivalent.**

5. **Option**: \( 3x + 3 = -x + 5 \)
- This also is not equivalent as simplified from before. **This is NOT equivalent.**

6. **Option**: \( 3x + 1 = -x + 5 \)
- This is also not equivalent. **This is NOT equivalent.**

Based on this evaluation, the correct options that Samuel can use are:

- \( 3x - 3 + 4 = 2 - x - 3 \)
- \( 3x + 1 = -x - 1 \)