Question

Select the equation that has infinitely many solutions.(1 point)
Responses

3x+5=3x−53 x plus 5 is equal to 3 x minus 5

3−3x=−3(x−1)3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren

3x=2x+13 x is equal to 2 x plus 1

4x+1=4(x+1)

Answer 1

To determine which equation has infinitely many solutions, we need to simplify each given equation and check if they are true for all values of \( x \).

1. **Equation 1**: \( 3x + 5 = 3x - 5 \)
\[
3x + 5 - 3x = -5 \implies 5 = -5 \quad \text{(not true)}
\]
This equation has no solutions.

2. **Equation 2**: \( 3 - 3x = -3(x - 1) \)
\[
3 - 3x = -3x + 3
\]
By adding \( 3x \) to both sides, we get:
\[
3 = 3 \quad \text{(true for any value of \( x \))}
\]
This equation has infinitely many solutions.

3. **Equation 3**: \( 3x = 2x + 1 \)
\[
3x - 2x = 1 \implies x = 1 \quad \text{(one solution)}
\]

4. **Equation 4**: \( 4x + 1 = 4(x + 1) \)
Expanding the right side:
\[
4x + 1 = 4x + 4
\]
Subtracting \( 4x \) from both sides gives:
\[
1 = 4 \quad \text{(not true)}
\]
This equation has no solutions.

The only equation with infinitely many solutions is **Equation 2**: \( 3 - 3x = -3(x - 1) \).