Question

match each equation with its number of solutions.
$9w - 2 = 9w + 2$
$9w - 2 = 8w + 2$
$9w - 2 = 9w - 2$
only one solution
no solution
infinitely many solutions

Explanation:

First Equation: \( 9w - 2 = 9w + 2 \)

Step 1: Subtract \( 9w \) from both sides

\( 9w - 9w - 2 = 9w - 9w + 2 \)
Simplify to get: \( -2 = 2 \)

Step 2: Analyze the result

The equation \( -2 = 2 \) is false, so this equation has no solution.

Second Equation: \( 9w - 2 = 8w + 2 \)

Step 1: Subtract \( 8w \) from both sides

\( 9w - 8w - 2 = 8w - 8w + 2 \)
Simplify to get: \( w - 2 = 2 \)

Step 2: Add 2 to both sides

\( w - 2 + 2 = 2 + 2 \)
Simplify to get: \( w = 4 \)
Since we found a unique value for \( w \), this equation has only one solution.

Third Equation: \( 9w - 2 = 9w - 2 \)

Step 1: Subtract \( 9w \) from both sides

\( 9w - 9w - 2 = 9w - 9w - 2 \)
Simplify to get: \( -2 = -2 \)

Step 2: Analyze the result

The equation \( -2 = -2 \) is always true, regardless of the value of \( w \). So this equation has infinitely many solutions.

Answer:

• \( 9w - 2 = 9w + 2 \): No solution
• \( 9w - 2 = 8w + 2 \): Only one solution
• \( 9w - 2 = 9w - 2 \): Infinitely many solutions