Question

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

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

Answer 1

Are u sure?

Answer 2

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**.

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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**.

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C. \( 10 - x = 25 \)

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

This equation has **one solution**.

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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**.

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To summarize:
- **A**: One solution
- **B**: No solutions
- **C**: One solution
- **D**: Infinitely many solutions

Answer 3

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

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

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

**This has one solution.**

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

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

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

This is a contradiction, meaning there are **no solutions.**

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**C.** \( 10 - x = 25 \)

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

**This has one solution.**

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

1. Expand the left side:
\[ 2x + 8 = 2x + 5 + 3 \]
\[ 2x + 8 = 2x + 8 \]

2. When simplified:
\[ 8 = 8 \]

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

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**Summary:**
- **A**: One solution.
- **B**: No solutions.
- **C**: One solution.
- **D**: Infinitely many solutions.

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