Question

Which 2 have infinite solutions?

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 infinite solutions, we need to simplify each equation and check for consistency.

1. **Equation:** \( 6(x - 9) = 6x - 54 + x \)

Simplifying the left side:
\[
6x - 54
\]
The right side simplifies to:
\[
6x - 54 + x = 7x - 54
\]
Thus, we have:
\[
6x - 54 = 7x - 54
\]
Subtracting \(6x\) from both sides:
\[
-54 = x - 54
\]
Adding \(54\) to both sides, we find:
\[
x = 0
\]
(This equation does not have infinite solutions.)

2. **Equation:** \( 5(x - 8) + 10 = 5(x + 2) \)

Simplifying the left side:
\[
5x - 40 + 10 = 5x - 30
\]
The right side simplifies to:
\[
5x + 10
\]
Thus, we have:
\[
5x - 30 = 5x + 10
\]
Subtracting \(5x\) from both sides gives:
\[
-30 = 10
\]
(This equation does not hold; it does not have infinite solutions.)

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

Simplifying the left side:
\[
14 - 8x
\]
The right side simplifies to:
\[
14 - 8x
\]
Thus, we have:
\[
14 - 8x = 14 - 8x
\]
This is true for all \(x\), so this equation has *infinite solutions*.

4. **Equation:** \( 4(x + 4) = 2(2x + 5) + 6 \)

Simplifying the left side:
\[
4x + 16
\]
The right side simplifies to:
\[
4x + 10 + 6 = 4x + 16
\]
Thus, we have:
\[
4x + 16 = 4x + 16
\]
This is also true for all \(x\), so this equation has *infinite solutions*.

5. **Equation:** \( 2x + 9 = 18 + 2x \)

Subtracting \(2x\) from both sides gives:
\[
9 = 18
\]
(This is not true; this equation does not have infinite solutions.)

**Conclusion:** The two equations that have infinite solutions are:

- \( 20 - 8x - 6 = 2(7- 4x) \)
- \( 4(x + 4) = 2(2x + 5) + 6 \)