Question

Which equation has infinitely many solutions?
1-2x=2(1-x)

Answer 1

To determine if the equation \( 1 - 2x = 2(1 - x) \) has infinitely many solutions, let's simplify the equation.

Start by expanding the right side:

\[
1 - 2x = 2 - 2x
\]

Next, rearrange both sides:

1. Subtract \( 2 \) from both sides:

\[
1 - 2 = 2 - 2x - 2x
\]
\[
-1 = 2 - 2x
\]

2. Now, subtract \( 1 \) from both sides:

\[
0 = 2 - 2x + 1
\]

This simplifies to:

\[
0 = 2 - 2x - 1
\]

Now combine like terms:

\[
0 = 1 - 2x
\]

3. Rearranging gives:

\[
2x = 1
\]

4. Divide by 2:

\[
x = \frac{1}{2}
\]

Since we have found a specific solution, \( x = \frac{1}{2} \), this equation only has one solution, not infinitely many.

Therefore, the equation \( 1 - 2x = 2(1 - x) \) does **not** have infinitely many solutions. It has exactly one solution:

\[
x = \frac{1}{2}
\]