Question
Which of the following systems describes a zero-solution?(1 point)
Responses
y=12x+6
and −6−2y=x
y equals Start Fraction 1 over 2 End Fraction x plus 6 and negative 6 minus 2 y equals x
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
Responses
y=12x+6
and −6−2y=x
y equals Start Fraction 1 over 2 End Fraction x plus 6 and negative 6 minus 2 y equals x
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
Answers
GPT-4o mini
To determine which system of equations describes a zero-solution (also known as an inconsistent system), we need to find a pair of equations that do not intersect, meaning they have no point in common.
Let's analyze each of the provided options:
1. **y = 12x + 6 and -6 - 2y = x**
- The first equation is a line with a slope of 12.
- The second equation can be rearranged to find y:
\[ -2y = x + 6 \]
\[ y = -\frac{1}{2}x - 3 \]
- These two lines have different slopes and will intersect at a point. So, it does not describe a zero-solution.
2. **6y = -12x + 18 and 8x + 4y = 2**
- The first equation can be expressed as:
\[ y = -2x + 3 \] (slope -2)
- The second equation can be expressed as:
\[ 4y = -8x + 2 \]
\[ y = -2x + \frac{1}{2} \] (slope -2)
- Both lines are parallel with the same slope but different y-intercepts. Thus, these equations will not intersect, leading to a zero-solution.
3. **y = x and y = -x**
- The first equation represents a line at 45 degrees through the origin, while the second line is at -45 degrees. They intersect at the origin (0, 0). Therefore, this does not describe a zero-solution.
4. **2y + x = 3 and 4x + 8y = 12**
- The second equation can be simplified to:
\[ 4y = -2x + 12 \]
\[ y = -\frac{1}{2}x + 3 \]
- The first equation can be expressed as:
\[ y = -\frac{1}{2}x + \frac{3}{2} \]
- Both lines have the same slope (-1/2) but different y-intercepts. Hence, they are parallel and do not intersect, leading to a zero-solution.
Based on the analysis, the systems that describe a zero-solution are:
- **Option 2: 6y = -12x + 18 and 8x + 4y = 2**
- **Option 4: 2y + x = 3 and 4x + 8y = 12**
If you must pick only one option, it is recommended to choose **Option 2** since it clearly demonstrates parallel lines.
Let's analyze each of the provided options:
1. **y = 12x + 6 and -6 - 2y = x**
- The first equation is a line with a slope of 12.
- The second equation can be rearranged to find y:
\[ -2y = x + 6 \]
\[ y = -\frac{1}{2}x - 3 \]
- These two lines have different slopes and will intersect at a point. So, it does not describe a zero-solution.
2. **6y = -12x + 18 and 8x + 4y = 2**
- The first equation can be expressed as:
\[ y = -2x + 3 \] (slope -2)
- The second equation can be expressed as:
\[ 4y = -8x + 2 \]
\[ y = -2x + \frac{1}{2} \] (slope -2)
- Both lines are parallel with the same slope but different y-intercepts. Thus, these equations will not intersect, leading to a zero-solution.
3. **y = x and y = -x**
- The first equation represents a line at 45 degrees through the origin, while the second line is at -45 degrees. They intersect at the origin (0, 0). Therefore, this does not describe a zero-solution.
4. **2y + x = 3 and 4x + 8y = 12**
- The second equation can be simplified to:
\[ 4y = -2x + 12 \]
\[ y = -\frac{1}{2}x + 3 \]
- The first equation can be expressed as:
\[ y = -\frac{1}{2}x + \frac{3}{2} \]
- Both lines have the same slope (-1/2) but different y-intercepts. Hence, they are parallel and do not intersect, leading to a zero-solution.
Based on the analysis, the systems that describe a zero-solution are:
- **Option 2: 6y = -12x + 18 and 8x + 4y = 2**
- **Option 4: 2y + x = 3 and 4x + 8y = 12**
If you must pick only one option, it is recommended to choose **Option 2** since it clearly demonstrates parallel lines.