Asked by TheOGBlurryface
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
A. x−7=7−x
B. 3(x+5)=3x+5
C. 10−x=25
D. 2(x+4)=2x+5+3
Answers
Answered by
TheOGBlurryface
Are u sure?
Answered by
GPT-4o mini
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:
- **A**: One solution
- **B**: No solutions
- **C**: One solution
- **D**: Infinitely many 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:
- **A**: One solution
- **B**: No solutions
- **C**: One solution
- **D**: Infinitely many solutions
Answered by
GPT-4o mini
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.**
---
**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.**
---
**C.** \( 10 - x = 25 \)
1. Rearranging gives:
\[ -x = 25 - 10 \]
\[ -x = 15 \]
\[ x = -15 \]
**This has one solution.**
---
**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**.
---
**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!
**A.** \( x - 7 = 7 - x \)
1. Rearranging gives:
\[ x + x = 7 + 7 \]
\[ 2x = 14 \]
\[ x = 7 \]
**This has one solution.**
---
**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.**
---
**C.** \( 10 - x = 25 \)
1. Rearranging gives:
\[ -x = 25 - 10 \]
\[ -x = 15 \]
\[ x = -15 \]
**This has one solution.**
---
**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**.
---
**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!