Question
Which of the following systems describes a zero-solution?(1 point)
Responses
2y+x=3
and 4x+8y=12
2 y plus x equals 3 and 4 x plus 8 y equals 12
y=x
and y=−x
y equals x and y equals negative 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=12x+6
and −6−2y=x
Responses
2y+x=3
and 4x+8y=12
2 y plus x equals 3 and 4 x plus 8 y equals 12
y=x
and y=−x
y equals x and y equals negative 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=12x+6
and −6−2y=x
Answers
GPT-4o mini
To find which of the given systems describes a zero-solution (meaning there are no solutions or the lines are parallel), we need to check whether the two equations in each set have the same slope.
1. **2y + x = 3** and **4x + 8y = 12**:
- Rearranging the first equation: \( y = -\frac{1}{2}x + \frac{3}{2} \) (Slope = -1/2)
- Rearranging the second equation: \( 8y = -4x + 12 \) ⇒ \( y = -\frac{1}{2}x + \frac{3}{2} \) (Slope = -1/2)
- Both equations have the same slope and the same y-intercept, so this is actually a dependent system (infinitely many solutions) not a zero-solution.
2. **y = x** and **y = -x**:
- Slopes: \( y = x \) has slope 1, \( y = -x \) has slope -1.
- These lines intersect at the origin (0,0), so there is one solution, not zero solutions.
3. **6y = -12x + 18** and **8x + 4y = 2**:
- Rearranging the first equation: \( y = -2x + 3 \) (Slope = -2)
- Rearranging the second equation: \( 4y = -8x + 2 \) ⇒ \( y = -2x + \frac{1}{2} \) (Slope = -2)
- Both equations have the same slope (-2) but different y-intercepts, meaning they are parallel and have no solution (zero-solution).
4. **y = 12x + 6** and **−6 − 2y = x**:
- Rearranging the second equation: \( -2y = x + 6 \) ⇒ \( y = -\frac{1}{2} x - 3 \) (Slope = -1/2).
- The first equation has a slope of 12 and the second has a slope of -1/2, meaning they are not parallel and thus will intersect at some point, yielding a solution.
Thus, the system that describes a zero-solution is:
**6y = -12x + 18 and 8x + 4y = 2**.
1. **2y + x = 3** and **4x + 8y = 12**:
- Rearranging the first equation: \( y = -\frac{1}{2}x + \frac{3}{2} \) (Slope = -1/2)
- Rearranging the second equation: \( 8y = -4x + 12 \) ⇒ \( y = -\frac{1}{2}x + \frac{3}{2} \) (Slope = -1/2)
- Both equations have the same slope and the same y-intercept, so this is actually a dependent system (infinitely many solutions) not a zero-solution.
2. **y = x** and **y = -x**:
- Slopes: \( y = x \) has slope 1, \( y = -x \) has slope -1.
- These lines intersect at the origin (0,0), so there is one solution, not zero solutions.
3. **6y = -12x + 18** and **8x + 4y = 2**:
- Rearranging the first equation: \( y = -2x + 3 \) (Slope = -2)
- Rearranging the second equation: \( 4y = -8x + 2 \) ⇒ \( y = -2x + \frac{1}{2} \) (Slope = -2)
- Both equations have the same slope (-2) but different y-intercepts, meaning they are parallel and have no solution (zero-solution).
4. **y = 12x + 6** and **−6 − 2y = x**:
- Rearranging the second equation: \( -2y = x + 6 \) ⇒ \( y = -\frac{1}{2} x - 3 \) (Slope = -1/2).
- The first equation has a slope of 12 and the second has a slope of -1/2, meaning they are not parallel and thus will intersect at some point, yielding a solution.
Thus, the system that describes a zero-solution is:
**6y = -12x + 18 and 8x + 4y = 2**.