Question
Which of the following systems describes a zero-solution?(1 point)
Responses
A) y=1/2x+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
B) 6y=−12x+18 and 8x+4y=2
6 y equals negative 12 x plus 18 and 8 x plus 4 y equals 2
C) y=x and y=−x
y equals x and y equals negative x
D) 2y+x=3 and 4x+8y=12
Responses
A) y=1/2x+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
B) 6y=−12x+18 and 8x+4y=2
6 y equals negative 12 x plus 18 and 8 x plus 4 y equals 2
C) y=x and y=−x
y equals x and y equals negative x
D) 2y+x=3 and 4x+8y=12
Answers
GPT-4o mini
To determine which system of equations describes a zero-solution, we need to check if the equations are inconsistent (i.e., they do not intersect at any point).
Let's analyze each option:
**A)**
1. \( y = \frac{1}{2}x + 6 \) (This is a line with a slope of \(\frac{1}{2}\) and y-intercept 6.)
2. \( -6 - 2y = x \) can be rewritten as \( x + 2y = -6 \) or \( 2y = -x - 6 \), which simplifies to \( y = -\frac{1}{2}x - 3 \) (This line has a slope of \(-\frac{1}{2}\) and y-intercept -3.)
These two lines have different slopes, so they will intersect at one point.
**B)**
1. \( 6y = -12x + 18 \) can be rewritten as \( y = -2x + 3 \) (This line has a slope of -2 and y-intercept of 3.)
2. \( 8x + 4y = 2 \) can be rewritten as \( 4y = -8x + 2 \) or \( y = -2x + \frac{1}{2} \) (This line also has a slope of -2 but a different y-intercept.)
Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect. Thus, this system has no solution.
**C)**
1. \( y = x \) (This is a line with slope 1.)
2. \( y = -x \) (This is a line with slope -1.)
These lines are perpendicular and will intersect at the origin (0,0). Therefore, this system has one solution.
**D)**
1. \( 2y + x = 3 \) can be rewritten as \( y = -\frac{1}{2}x + \frac{3}{2} \) (This line has a slope of -\(\frac{1}{2}\) and y-intercept \(\frac{3}{2}\).)
2. \( 4x + 8y = 12 \) simplifies to \( 8y = -4x + 12 \), or \( y = -\frac{1}{2}x + \frac{3}{2} \) (This line is identical to the first equation).
Since they are the same line, they have infinitely many solutions.
### Conclusion:
The system that describes a **zero-solution** is **B)**: \( 6y = -12x + 18 \) and \( 8x + 4y = 2 \) because these lines are parallel and do not intersect.
Let's analyze each option:
**A)**
1. \( y = \frac{1}{2}x + 6 \) (This is a line with a slope of \(\frac{1}{2}\) and y-intercept 6.)
2. \( -6 - 2y = x \) can be rewritten as \( x + 2y = -6 \) or \( 2y = -x - 6 \), which simplifies to \( y = -\frac{1}{2}x - 3 \) (This line has a slope of \(-\frac{1}{2}\) and y-intercept -3.)
These two lines have different slopes, so they will intersect at one point.
**B)**
1. \( 6y = -12x + 18 \) can be rewritten as \( y = -2x + 3 \) (This line has a slope of -2 and y-intercept of 3.)
2. \( 8x + 4y = 2 \) can be rewritten as \( 4y = -8x + 2 \) or \( y = -2x + \frac{1}{2} \) (This line also has a slope of -2 but a different y-intercept.)
Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect. Thus, this system has no solution.
**C)**
1. \( y = x \) (This is a line with slope 1.)
2. \( y = -x \) (This is a line with slope -1.)
These lines are perpendicular and will intersect at the origin (0,0). Therefore, this system has one solution.
**D)**
1. \( 2y + x = 3 \) can be rewritten as \( y = -\frac{1}{2}x + \frac{3}{2} \) (This line has a slope of -\(\frac{1}{2}\) and y-intercept \(\frac{3}{2}\).)
2. \( 4x + 8y = 12 \) simplifies to \( 8y = -4x + 12 \), or \( y = -\frac{1}{2}x + \frac{3}{2} \) (This line is identical to the first equation).
Since they are the same line, they have infinitely many solutions.
### Conclusion:
The system that describes a **zero-solution** is **B)**: \( 6y = -12x + 18 \) and \( 8x + 4y = 2 \) because these lines are parallel and do not intersect.