Question

Which equation has infinitely many solutions?
4x+3/3 = 4x + 5
10x+8/2 = 5x+4
8x-5/3 = 2x-2
8x-5/2 = 4x-6

Answer 1

To determine which equation has infinitely many solutions, we'll solve each equation and see if it leads to a true statement for all values of \(x\).

1. **Equation 1:**
\[
\frac{4x + 3}{3} = 4x + 5
\]
Multiply both sides by 3:
\[
4x + 3 = 12x + 15
\]
Rearranging gives:
\[
4x - 12x = 15 - 3
\]
\[
-8x = 12 \quad \Rightarrow \quad x = -\frac{12}{8} = -\frac{3}{2}
\]
This equation has one solution: \(x = -\frac{3}{2}\).

2. **Equation 2:**
\[
\frac{10x + 8}{2} = 5x + 4
\]
Multiply both sides by 2:
\[
10x + 8 = 10x + 8
\]
This simplifies to:
\[
0 = 0
\]
Since this is always true for any value of \(x\), this equation has infinitely many solutions.

3. **Equation 3:**
\[
\frac{8x - 5}{3} = 2x - 2
\]
Multiply both sides by 3:
\[
8x - 5 = 6x - 6
\]
Rearranging gives:
\[
8x - 6x = -6 + 5
\]
\[
2x = -1 \quad \Rightarrow \quad x = -\frac{1}{2}
\]
This equation has one solution: \(x = -\frac{1}{2}\).

4. **Equation 4:**
\[
\frac{8x - 5}{2} = 4x - 6
\]
Multiply both sides by 2:
\[
8x - 5 = 8x - 12
\]
Rearranging gives:
\[
-5 = -12
\]
This is a false statement, so this equation has no solutions.

**Conclusion:**
The equation that has infinitely many solutions is:
\[
\frac{10x + 8}{2} = 5x + 4
\]