Question
Match the system of equations to the number of solutions it has. (4 points)
COURSE TOOLS
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y = 5x + 7 and 3y15z 18
Item 2
3x + 6 = y and y = - 1/3 * x - 4
Item 3
y = 2/3 * x - an * dy = 2/3 * x - 2
Item 4
Item 5
x - 2y = 6 and 3x - 6y = 18
Item 6
One Solution
Item 1
:: Infinite solutions
:: No solutions
COURSE TOOLS
<
<
y = 5x + 7 and 3y15z 18
Item 2
3x + 6 = y and y = - 1/3 * x - 4
Item 3
y = 2/3 * x - an * dy = 2/3 * x - 2
Item 4
Item 5
x - 2y = 6 and 3x - 6y = 18
Item 6
One Solution
Item 1
:: Infinite solutions
:: No solutions
Answers
GPT-4o mini
To determine the number of solutions for each system of equations, let's analyze them one by one:
**Item 1:**
Equations: \( y = 5x + 7 \) and \( 3y = 15z + 18 \).
Analysis: This is a linear equation in two variables \( (x, y) \) and a different equation in \( y \) and \( z \). Since there is no way to equate these two or graph them in a consistent manner, we need more information. However, this implies the system does not have a solution relating the two variables, so we would classify this as:
- **No solutions**
---
**Item 2:**
Equations: \( 3x + 6 = y \) and \( y = -\frac{1}{3}x - 4 \).
Rearranging both equations gives:
1. \( y = 3x + 6 \)
2. \( y = -\frac{1}{3}x - 4 \)
The slopes of these two lines are different (3 vs. -1/3), so they will intersect at one point. Thus, this system has:
- **One Solution**
---
**Item 3:**
Equations: \( y = \frac{2}{3}x - an \) and \( y = \frac{2}{3}x - 2 \).
Since the slopes of both lines are the same (\(\frac{2}{3}\)), they are parallel lines. Therefore, this system has:
- **No solutions**
---
**Item 4:**
Equations: \( x - 2y = 6 \) and \( 3x - 6y = 18 \).
Rewriting the second equation gives:
\( 3(x - 2y) = 3(6) \), which means both equations represent the same line. Thus, this system has:
- **Infinite solutions**
---
Based on this analysis, we can summarize the results as:
- **Item 1:** No solutions
- **Item 2:** One solution
- **Item 3:** No solutions
- **Item 4:** Infinite solutions
Hopefully, this breakdown helps clarify the relationships between the equations and the solutions!
**Item 1:**
Equations: \( y = 5x + 7 \) and \( 3y = 15z + 18 \).
Analysis: This is a linear equation in two variables \( (x, y) \) and a different equation in \( y \) and \( z \). Since there is no way to equate these two or graph them in a consistent manner, we need more information. However, this implies the system does not have a solution relating the two variables, so we would classify this as:
- **No solutions**
---
**Item 2:**
Equations: \( 3x + 6 = y \) and \( y = -\frac{1}{3}x - 4 \).
Rearranging both equations gives:
1. \( y = 3x + 6 \)
2. \( y = -\frac{1}{3}x - 4 \)
The slopes of these two lines are different (3 vs. -1/3), so they will intersect at one point. Thus, this system has:
- **One Solution**
---
**Item 3:**
Equations: \( y = \frac{2}{3}x - an \) and \( y = \frac{2}{3}x - 2 \).
Since the slopes of both lines are the same (\(\frac{2}{3}\)), they are parallel lines. Therefore, this system has:
- **No solutions**
---
**Item 4:**
Equations: \( x - 2y = 6 \) and \( 3x - 6y = 18 \).
Rewriting the second equation gives:
\( 3(x - 2y) = 3(6) \), which means both equations represent the same line. Thus, this system has:
- **Infinite solutions**
---
Based on this analysis, we can summarize the results as:
- **Item 1:** No solutions
- **Item 2:** One solution
- **Item 3:** No solutions
- **Item 4:** Infinite solutions
Hopefully, this breakdown helps clarify the relationships between the equations and the solutions!