Question

Which linear equations have one solution? Check all that apply.

5x – 1 = 3(x + 11)
4(x – 2) + 4x = 8(x – 9)
4(x – 6) + 4 = 2(x – 3)
2(x – 4) = 5(x – 3) + 3
2(x – 1) + 3x = 5(x – 2) + 3

Answer 1

To determine which linear equations have one solution, we need to simplify each equation to see if they are consistent and can be solved to yield a unique solution. Let's go through the equations one by one:

1. **5x – 1 = 3(x + 11)**
- Simplifying:
\( 5x - 1 = 3x + 33 \)
\( 5x - 3x = 33 + 1 \)
\( 2x = 34 \)
\( x = 17 \)
- Result: **One solution** (x = 17)

2. **4(x – 2) + 4x = 8(x – 9)**
- Simplifying:
\( 4x - 8 + 4x = 8x - 72 \)
\( 8x - 8 = 8x - 72 \)
Subtracting \( 8x \) from both sides:
\( -8 = -72 \)
- Result: This is a **contradiction**, so there are **no solutions**.

3. **4(x – 6) + 4 = 2(x – 3)**
- Simplifying:
\( 4x - 24 + 4 = 2x - 6 \)
\( 4x - 20 = 2x - 6 \)
\( 4x - 2x = -6 + 20 \)
\( 2x = 14 \)
\( x = 7 \)
- Result: **One solution** (x = 7)

4. **2(x – 4) = 5(x – 3) + 3**
- Simplifying:
\( 2x - 8 = 5x - 15 + 3 \)
\( 2x - 8 = 5x - 12 \)
\( 2x - 5x = -12 + 8 \)
\( -3x = -4 \)
\( x = \frac{4}{3} \)
- Result: **One solution** (x = 4/3)

5. **2(x – 1) + 3x = 5(x – 2) + 3**
- Simplifying:
\( 2x - 2 + 3x = 5x - 10 + 3 \)
\( 5x - 2 = 5x - 7 \)
Subtracting \( 5x \) from both sides:
\( -2 = -7 \)
- Result: This is a **contradiction**, so there are **no solutions**.

### Summary:
- Equations with **one solution**:
- **5x – 1 = 3(x + 11)**
- **4(x – 6) + 4 = 2(x – 3)**
- **2(x – 4) = 5(x – 3) + 3**

- Equations with **no solutions**:
- **4(x – 2) + 4x = 8(x – 9)**
- **2(x – 1) + 3x = 5(x – 2) + 3**