Question

Which equations have infinitely many solutions? Select two that apply.

6(x - 9) = 6x - 54 + x

5(x - 8) + 10 = 5(x + 2)

20 - 8x - 6 = 2(7 - 4x)

4(x + 4) = 2(2x + 5) + 6

2x + 9 = 18 + 2x

Answer 1

To determine which equations have infinitely many solutions, we need to simplify and analyze each equation.

1. **Equation: \( 6(x - 9) = 6x - 54 + x \)**
- Simplifying the left side: \( 6(x - 9) = 6x - 54 \)
- Right side: \( 6x - 54 + x = 7x - 54 \)
- The equation simplifies to \( 6x - 54 = 7x - 54 \)
- Rearranging: \( 6x - 7x = -54 + 54 \) ➔ \( -x = 0 \) ➔ \( x = 0 \) (One solution)

2. **Equation: \( 5(x - 8) + 10 = 5(x + 2) \)**
- Left side: \( 5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30 \)
- Right side: \( 5(x + 2) = 5x + 10 \)
- The equation simplifies to \( 5x - 30 = 5x + 10 \)
- Rearranging: \( 5x - 5x = 10 + 30 \) ➔ \( 0 = 40 \) (Inconsistent, no solutions)

3. **Equation: \( 20 - 8x - 6 = 2(7 - 4x) \)**
- Left side: \( 20 - 8x - 6 = 14 - 8x \)
- Right side: \( 2(7 - 4x) = 14 - 8x \)
- The equation simplifies to \( 14 - 8x = 14 - 8x \) (True for all \( x \))
- This means there are infinitely many solutions.

4. **Equation: \( 4(x + 4) = 2(2x + 5) + 6 \)**
- Left side: \( 4(x + 4) = 4x + 16 \)
- Right side: \( 2(2x + 5) + 6 = 4x + 10 + 6 = 4x + 16 \)
- The equation simplifies to \( 4x + 16 = 4x + 16 \) (True for all \( x \))
- This means there are infinitely many solutions.

5. **Equation: \( 2x + 9 = 18 + 2x \)**
- Rearranging gives \( 2x + 9 - 2x = 18 \) ➔ \( 9 = 18 \) (Inconsistent, no solutions)

Based on this analysis, the two equations that have infinitely many solutions are:

- **Equation 3:** \( 20 - 8x - 6 = 2(7 - 4x) \)
- **Equation 4:** \( 4(x + 4) = 2(2x + 5) + 6 \)