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

Question

A. x−7=7−x
B. 3(x+5)=3x+5
C. 10−x=25
D. 2(x+4)=2x+5+3

GPT-4o mini · Answer

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

A. \( x - 7 = 7 - x \)

Combine like terms: \[ x + x = 7 + 7 \] \[ 2x = 14 \] \[ x = 7 \]

This equation has one solution.

B. \( 3(x + 5) = 3x + 5 \)

Expand the left side: \[ 3x + 15 = 3x + 5 \]

Subtract \( 3x \) from both sides: \[ 15 = 5 \]

This is a false statement, indicating that there is no solution. So this equation has no solutions.

C. \( 10 - x = 25 \)

Rearranging gives: \[ -x = 25 - 10 \] \[ -x = 15 \] \[ x = -15 \]

This equation has one solution.

D. \( 2(x + 4) = 2x + 5 + 3 \)

Expand the left side: \[ 2x + 8 = 2x + 8 \]

Subtract \( 2x \) from both sides: \[ 8 = 8 \]

This is a true statement for all values of \( x \). Therefore, this equation has infinitely many solutions.

To summarize:

TheOGBlurryface · Answer

Are u sure?

GPT-4o mini · Answer

Let’s go through each equation again to ensure accuracy:

A. \( x - 7 = 7 - x \)

Rearranging gives: \[ x + x = 7 + 7 \] \[ 2x = 14 \] \[ x = 7 \]

This has one solution.

B. \( 3(x + 5) = 3x + 5 \)

Expand the left side: \[ 3x + 15 = 3x + 5 \]
Subtract \( 3x \) from both sides: \[ 15 = 5 \]

This is a contradiction, meaning there are no solutions.

C. \( 10 - x = 25 \)

Rearranging gives: \[ -x = 25 - 10 \] \[ -x = 15 \] \[ x = -15 \]

This has one solution.

D. \( 2(x + 4) = 2x + 5 + 3 \)

Expand the left side: \[ 2x + 8 = 2x + 5 + 3 \] \[ 2x + 8 = 2x + 8 \]
When simplified: \[ 8 = 8 \]

This is always true for all values of \( x\), indicating infinitely many solutions.

Summary:

This confirms the previous analysis and conclusions. If you have further questions or need additional clarification, feel free to ask!