Which system makes the statement true? "The system has infinitely many solutions" -5x+ 4y = 5 -15x+ 12y = 15 I. -5x+ 4y = 5 II -15x+ 12y = 5 -5x+ 4y = 5 III -15x+ 4y = 15
Which system makes the statement true? "The system has infinitely many solutions" -5x+ 4y = 5 -15x+ 12y = 15 I. -5x+ 4y = 5 II -15x+ 12y = 5 -5x+ 4y = 5 III -15x+ 4y = 15
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Which system makes the statement true?
**The system has infinitely many solutions**
**Systems of Equations:**
- \(-5x + 4y = 5\)
**System I:**
- \[-15x + 12y = 15\]
**System II:**
- \[-15x + 12y = 5\]
**System III:**
- \[-15x + 4y = 15\]
**Options:**
1. ( ) III
2. ( ) I
3. ( ) I and II
4. ( ) II
### Explanation
To determine which system of equations has infinitely many solutions, we analyze each system to see if they are consistent and dependent. This occurs when the equations are essentially the same, differing only by a multiplicative constant.
- **System I**:
\[-15x + 12y = 15\]
If you multiply the first equation \(-5x + 4y = 5\) by 3, you get \((-5x \times 3) + (4y \times 3) = 5 \times 3\), which simplifies to \(-15x + 12y = 15\). Thus, System I has infinitely many solutions as they are just multiples of each other.
- **System II**:
\[-15x + 12y = 5\]
Multiplying the first equation \(-5x + 4y = 5\) by 3, as above, gives \(-15x + 12y = 15\). The given equation for System II is not the same as this result, hence they do not have infinitely many solutions together.
- **System III**:
\[-15x + 4y = 15\]
Comparing this with \(-5x + 4y = 5\), no constant multiplier converts \(-5x + 4y = 5\) into \(-15x + 4y = 15\). Therefore, System III does not have infinitely many solutions.
**The correct answer is:**
( ) I](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F902e7e06-dd9d-47fe-961b-7a721406cdf1%2F8014c1a5-192a-4b02-8279-faa0552386c5%2Fg25hjxw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Which system makes the statement true?
**The system has infinitely many solutions**
**Systems of Equations:**
- \(-5x + 4y = 5\)
**System I:**
- \[-15x + 12y = 15\]
**System II:**
- \[-15x + 12y = 5\]
**System III:**
- \[-15x + 4y = 15\]
**Options:**
1. ( ) III
2. ( ) I
3. ( ) I and II
4. ( ) II
### Explanation
To determine which system of equations has infinitely many solutions, we analyze each system to see if they are consistent and dependent. This occurs when the equations are essentially the same, differing only by a multiplicative constant.
- **System I**:
\[-15x + 12y = 15\]
If you multiply the first equation \(-5x + 4y = 5\) by 3, you get \((-5x \times 3) + (4y \times 3) = 5 \times 3\), which simplifies to \(-15x + 12y = 15\). Thus, System I has infinitely many solutions as they are just multiples of each other.
- **System II**:
\[-15x + 12y = 5\]
Multiplying the first equation \(-5x + 4y = 5\) by 3, as above, gives \(-15x + 12y = 15\). The given equation for System II is not the same as this result, hence they do not have infinitely many solutions together.
- **System III**:
\[-15x + 4y = 15\]
Comparing this with \(-5x + 4y = 5\), no constant multiplier converts \(-5x + 4y = 5\) into \(-15x + 4y = 15\). Therefore, System III does not have infinitely many solutions.
**The correct answer is:**
( ) I
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