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

Answer 1

To determine which of the given systems describes a zero-solution (i.e., a system with no solutions, where the lines are parallel), we can analyze each pair of equations. A system has no solutions if the lines are parallel, which occurs when they have the same slope but different y-intercepts.

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

Rearranging the second equation:
\[
6y = -3x + 5 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{5}{6}
\]
The slopes are -12 and -1/2. **Not parallel.**

2. **System 2:**
\( y = 12x + 6 \) and \( -6 - 2y = x \)

Rearranging the second equation:
\[
-2y = x + 6 \quad \Rightarrow \quad 2y = -x - 6 \quad \Rightarrow \quad y = -\frac{1}{2}x - 3
\]
The slopes are 12 and -1/2. **Not parallel.**

3. **System 3:**
\( y = x \) and \( y = -x \)

The slopes are 1 and -1. **Not parallel.**

4. **System 4:**
\( 2y + x = 3 \) and \( 4x + 8y = 12 \)

Rearranging the first equation:
\[
2y = -x + 3 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{3}{2}
\]
The second equation can be simplified:
\[
4x + 8y = 12 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{3}{2}
\]
The slopes are both -1/2, and since they have the same y-intercept, they are actually the same line. **Not zero-solution.**

After analyzing all options, it appears that none of these systems have a zero-solution. If we were to choose the answer that is closest to a zero-solution in this context, we would need to identify a pair of equations with the same slope but different y-intercepts—which none of these fulfill. Therefore, none of the systems listed describes a zero-solution (no solutions).