Question
Which of the following systems describes a zero-solution?(1 point)
Responses
6y=−12x+18
and 8x+4y=2
6 y equals negative 12 x plus 18 and 8 x plus 4 y equals 2
y=x
and y=−x
y equals x and y equals negative x
2y+x=3
and 4x+8y=12
2 y plus x equals 3 and 4 x plus 8 y equals 12
y=12x+6
and −6−2y=x
Responses
6y=−12x+18
and 8x+4y=2
6 y equals negative 12 x plus 18 and 8 x plus 4 y equals 2
y=x
and y=−x
y equals x and y equals negative x
2y+x=3
and 4x+8y=12
2 y plus x equals 3 and 4 x plus 8 y equals 12
y=12x+6
and −6−2y=x
Answers
GPT-4o mini
A zero-solution in a system of equations means that there are no values of \( x \) and \( y \) that satisfy both equations simultaneously (the lines represented by the equations are parallel and never intersect).
Let's analyze each of the provided pairs of equations:
1. **6y = -12x + 18 and 8x + 4y = 2**
- The first equation can be rearranged into slope-intercept form: \( y = -2x + 3 \) (slope = -2).
- The second equation: \( 4y = -8x + 2 \rightarrow y = -2x + \frac{1}{2} \) (slope = -2).
- Since both equations have the same slope, they are parallel, and there is no intersection. **This system has a zero-solution.**
2. **y = x and y = -x**
- These equations represent two lines with slopes 1 and -1, which intersect at the origin (0,0). **This system has a solution.**
3. **2y + x = 3 and 4x + 8y = 12**
- The second equation can be rewritten as \( 8y = -4x + 12 \rightarrow y = -\frac{1}{2}x + \frac{3}{2} \) (slope = -1/2).
- The first equation can be rewritten as \( 2y = -x + 3 \rightarrow y = -\frac{1}{2}x + \frac{3}{2} \) (slope = -1/2).
- Both equations have the same slope and y-intercept and represent the same line. **This system has infinitely many solutions, not a zero-solution.**
4. **y = 12x + 6 and -6 - 2y = x**
- The second equation can be rearranged to slope-intercept form: \( -2y = x + 6 \rightarrow y = -\frac{1}{2}x - 3 \) (slope = -1/2).
- The first equation has a slope of 12, which is different from -1/2. Therefore, the lines will intersect. **This system has a solution.**
Thus, the only system from the options that describes a zero-solution is:
**6y = -12x + 18 and 8x + 4y = 2**
Let's analyze each of the provided pairs of equations:
1. **6y = -12x + 18 and 8x + 4y = 2**
- The first equation can be rearranged into slope-intercept form: \( y = -2x + 3 \) (slope = -2).
- The second equation: \( 4y = -8x + 2 \rightarrow y = -2x + \frac{1}{2} \) (slope = -2).
- Since both equations have the same slope, they are parallel, and there is no intersection. **This system has a zero-solution.**
2. **y = x and y = -x**
- These equations represent two lines with slopes 1 and -1, which intersect at the origin (0,0). **This system has a solution.**
3. **2y + x = 3 and 4x + 8y = 12**
- The second equation can be rewritten as \( 8y = -4x + 12 \rightarrow y = -\frac{1}{2}x + \frac{3}{2} \) (slope = -1/2).
- The first equation can be rewritten as \( 2y = -x + 3 \rightarrow y = -\frac{1}{2}x + \frac{3}{2} \) (slope = -1/2).
- Both equations have the same slope and y-intercept and represent the same line. **This system has infinitely many solutions, not a zero-solution.**
4. **y = 12x + 6 and -6 - 2y = x**
- The second equation can be rearranged to slope-intercept form: \( -2y = x + 6 \rightarrow y = -\frac{1}{2}x - 3 \) (slope = -1/2).
- The first equation has a slope of 12, which is different from -1/2. Therefore, the lines will intersect. **This system has a solution.**
Thus, the only system from the options that describes a zero-solution is:
**6y = -12x + 18 and 8x + 4y = 2**