Question

Which of the following systems describes a zero-solution?(1 point)
Responses

y=−12x+2 and 3x+6y=5
y=−12x+2 and 3x+6y=5

y=12x+6 and −6−2y=x
y=12x+6 and −6−2y=x

y=x and y=−x
y=x and y=−x

2y+x=3 and 4x+8y=12
2y+x=3 and 4x+8y=12

Answer 1

To determine which pair of systems has a zero-solution (meaning there are no solutions or the lines do not intersect), we need to analyze the given systems of equations and see if they represent parallel lines or inconsistent equations.

1. **First pair:**
\(y = -12x + 2\)
\(3x + 6y = 5\)

Converting the second equation to the slope-intercept form:
\(6y = -3x + 5\)
\(y = -\frac{1}{2}x + \frac{5}{6}\)
The slopes are different (-12 and -1/2), so they intersect. Thus, this system has solutions.

2. **Second pair:**
\(y = 12x + 6\)
\(-6 - 2y = x\)
or rearranged:
\(2y = -x - 6\)
\(y = -\frac{1}{2}x - 3\)

The slopes are different (12 and -1/2), so they intersect. Thus, this system has solutions.

3. **Third pair:**
\(y = x\)
\(y = -x\)

These lines intersect at the origin (0,0). Thus, this system has solutions.

4. **Fourth pair:**
\(2y + x = 3\)
Rearranged:
\(2y = -x + 3\)
\(y = -\frac{1}{2}x + \frac{3}{2}\)

\(4x + 8y = 12\) can be simplified:
\(8y = -4x + 12\)
\(y = -\frac{1}{2}x + \frac{3}{2}\)

Both equations represent the same line, which means there are infinitely many solutions. Therefore, this does not describe a zero-solution.

Given these analyses, none of the systems describes a zero-solution. They all either have a unique solution or infinitely many solutions.