Asked by AYD3N
The point (5,-3) is a solution to which system of equations? (1 point)
A. y=x-8 2x+y=7
B. y=2/3x+6 3y+6x-18=0
C. y=x+2 y=x+5
D. y=-1/2x+6 y=3x-1
A. y=x-8 2x+y=7
B. y=2/3x+6 3y+6x-18=0
C. y=x+2 y=x+5
D. y=-1/2x+6 y=3x-1
Answers
Answered by
AYD3N
How many solutions does the system of equations have?
y=-2x+12
4y+x-20=0 (1 point)
A. Infinite solutions
B. One solution
C. Two solutions
D. No solution
y=-2x+12
4y+x-20=0 (1 point)
A. Infinite solutions
B. One solution
C. Two solutions
D. No solution
Answered by
GPT-4o mini
To determine which system of equations has the point (5, -3) as a solution, we need to substitute \( x = 5 \) and \( y = -3 \) into each equation in the systems provided and see which system holds true for both equations.
### Option A:
1. **Equation 1:** \( y = x - 8 \)
Substitute \( x = 5 \):
\( y = 5 - 8 = -3 \) (True)
2. **Equation 2:** \( 2x + y = 7 \)
Substitute \( x = 5 \) and \( y = -3 \):
\( 2(5) + (-3) = 10 - 3 = 7 \) (True)
Both equations in Option A are satisfied.
### Option B:
1. **Equation 1:** \( y = \frac{2}{3}x + 6 \)
Substitute \( x = 5 \):
\( y = \frac{2}{3}(5) + 6 = \frac{10}{3} + 6 = \frac{10}{3} + \frac{18}{3} = \frac{28}{3} \) (Not true, since -3 is not equal to \(\frac{28}{3}\))
Since the first equation is not satisfied, we can discard this option.
### Option C:
1. **Equation 1:** \( y = x + 2 \)
Substitute \( x = 5 \):
\( y = 5 + 2 = 7 \) (Not true since -3 is not equal to 7)
Again, since the first equation is not satisfied, we can discard this option.
### Option D:
1. **Equation 1:** \( y = -\frac{1}{2}x + 6 \)
Substitute \( x = 5 \):
\( y = -\frac{1}{2}(5) + 6 = -\frac{5}{2} + 6 = -\frac{5}{2} + \frac{12}{2} = \frac{7}{2} \) (Not true since -3 is not equal to \(\frac{7}{2}\))
Since the first equation is not satisfied, we can discard this option.
### Conclusion:
After checking all options, the only system that satisfies the point (5, -3) is:
**Option A:** \( y = x - 8 \) and \( 2x + y = 7 \)
### Option A:
1. **Equation 1:** \( y = x - 8 \)
Substitute \( x = 5 \):
\( y = 5 - 8 = -3 \) (True)
2. **Equation 2:** \( 2x + y = 7 \)
Substitute \( x = 5 \) and \( y = -3 \):
\( 2(5) + (-3) = 10 - 3 = 7 \) (True)
Both equations in Option A are satisfied.
### Option B:
1. **Equation 1:** \( y = \frac{2}{3}x + 6 \)
Substitute \( x = 5 \):
\( y = \frac{2}{3}(5) + 6 = \frac{10}{3} + 6 = \frac{10}{3} + \frac{18}{3} = \frac{28}{3} \) (Not true, since -3 is not equal to \(\frac{28}{3}\))
Since the first equation is not satisfied, we can discard this option.
### Option C:
1. **Equation 1:** \( y = x + 2 \)
Substitute \( x = 5 \):
\( y = 5 + 2 = 7 \) (Not true since -3 is not equal to 7)
Again, since the first equation is not satisfied, we can discard this option.
### Option D:
1. **Equation 1:** \( y = -\frac{1}{2}x + 6 \)
Substitute \( x = 5 \):
\( y = -\frac{1}{2}(5) + 6 = -\frac{5}{2} + 6 = -\frac{5}{2} + \frac{12}{2} = \frac{7}{2} \) (Not true since -3 is not equal to \(\frac{7}{2}\))
Since the first equation is not satisfied, we can discard this option.
### Conclusion:
After checking all options, the only system that satisfies the point (5, -3) is:
**Option A:** \( y = x - 8 \) and \( 2x + y = 7 \)
Answered by
GPT-4o mini
To determine how many solutions the system of equations has, we can analyze the two equations given:
1. **Equation 1:** \( y = -2x + 12 \)
2. **Equation 2:** \( 4y + x - 20 = 0 \)
First, let's rewrite Equation 2 in slope-intercept form (i.e., \(y = mx + b\)):
Starting with:
\[ 4y + x - 20 = 0 \]
We can isolate \(y\):
\[ 4y = -x + 20 \]
Now, divide by 4:
\[ y = -\frac{1}{4}x + 5 \]
Now, we have both equations in the form of \(y = mx + b\):
1. \( y = -2x + 12 \) (slope = -2)
2. \( y = -\frac{1}{4}x + 5 \) (slope = -1/4)
### Analysis
- The slopes of the two lines are different (-2 and -1/4).
- Since the slopes are not equal, the lines will intersect at one point.
### Conclusion
The system of equations has **one solution**.
The correct answer is:
**B. One solution**.
1. **Equation 1:** \( y = -2x + 12 \)
2. **Equation 2:** \( 4y + x - 20 = 0 \)
First, let's rewrite Equation 2 in slope-intercept form (i.e., \(y = mx + b\)):
Starting with:
\[ 4y + x - 20 = 0 \]
We can isolate \(y\):
\[ 4y = -x + 20 \]
Now, divide by 4:
\[ y = -\frac{1}{4}x + 5 \]
Now, we have both equations in the form of \(y = mx + b\):
1. \( y = -2x + 12 \) (slope = -2)
2. \( y = -\frac{1}{4}x + 5 \) (slope = -1/4)
### Analysis
- The slopes of the two lines are different (-2 and -1/4).
- Since the slopes are not equal, the lines will intersect at one point.
### Conclusion
The system of equations has **one solution**.
The correct answer is:
**B. One solution**.
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