Match the description of the one variable equation with the number of solutions it will have

Question

x−7=7−x

3(x+5)=3x+5

10−x=25

2(x+4)=2x+5+3

Answer 1

Let's analyze each equation one by one to determine the number of solutions for each.

1. Equation: \( x - 7 = 7 - x \)

   • Rearranging the equation: \( x + x = 7 + 7 \) leads to \( 2x = 14 \), so \( x = 7 \).
   • Number of Solutions: 1

2. Equation: \( 3(x + 5) = 3x + 5 \)

   • Distributing \( 3 \) on the left side gives \( 3x + 15 = 3x + 5 \).
   • Subtracting \( 3x \) from both sides results in \( 15 = 5 \), which is a false statement.
   • Number of Solutions: 0

3. Equation: \( 10 - x = 25 \)

   • Rearranging gives \( -x = 25 - 10 \), so \( -x = 15 \) or \( x = -15 \).
   • Number of Solutions: 1

4. Equation: \( 2(x + 4) = 2x + 5 + 3 \)

   • Distributing \( 2 \) on the left side gives \( 2x + 8 = 2x + 8 \).
   • This simplifies to \( 0 = 0 \), which is always true, meaning there are infinitely many solutions.
   • Number of Solutions: Infinite

In summary:

• \( x - 7 = 7 - x \): 1 solution
• \( 3(x + 5) = 3x + 5 \): 0 solutions
• \( 10 - x = 25 \): 1 solution
• \( 2(x + 4) = 2x + 5 + 3 \): Infinite solutions