Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
What value for a will result into a equation with infintely many solutions?
![**Question 6:**
What value for \( a \) will result in an equation with infinitely many solutions?
Given equation:
\[ a(x + 5) = 3x + 15 \]
**Detailed Explanation:**
To find the value of \( a \) that results in infinitely many solutions, set the expressions equal by comparing the coefficients and constants once the equation is simplified.
### Steps to Solve:
1. **Distribute \( a \) in the expression:**
\[
a(x + 5) = ax + 5a
\]
2. **Set the equations equal:**
\[
ax + 5a = 3x + 15
\]
3. **Compare coefficients and constants:**
- Coefficient of \( x \): \( a = 3 \)
- Constant term: \( 5a = 15 \)
4. **Solve for \( a \) in the constant equation:**
\[
5a = 15 \implies a = \frac{15}{5} = 3
\]
Since both conditions agree, \( a = 3 \) is the correct value for infinitely many solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbac3ff1-75d8-4334-8c05-508631b41bdb%2F33677ae6-910e-4060-a331-a7df773f2364%2Fo7mw8fs_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 6:**
What value for \( a \) will result in an equation with infinitely many solutions?
Given equation:
\[ a(x + 5) = 3x + 15 \]
**Detailed Explanation:**
To find the value of \( a \) that results in infinitely many solutions, set the expressions equal by comparing the coefficients and constants once the equation is simplified.
### Steps to Solve:
1. **Distribute \( a \) in the expression:**
\[
a(x + 5) = ax + 5a
\]
2. **Set the equations equal:**
\[
ax + 5a = 3x + 15
\]
3. **Compare coefficients and constants:**
- Coefficient of \( x \): \( a = 3 \)
- Constant term: \( 5a = 15 \)
4. **Solve for \( a \) in the constant equation:**
\[
5a = 15 \implies a = \frac{15}{5} = 3
\]
Since both conditions agree, \( a = 3 \) is the correct value for infinitely many solutions.
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