Question

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

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

y=1/2x+6 and −6−2y=x

6y=−1/2x+18 and 8x+4y=2

y=x and y=−x

Answer 1

To determine which of the given systems has a zero solution, we need to check if the equations in each system intersect at the origin (0, 0), or if they yield a result where no solution exists. We will analyze each option:

1. **System 1:**
- \(2y + x = 3\)
- \(4x + 8y = 12\)

We can simplify the second equation:
\((4x + 8y = 12)\) can be divided by 4, giving \(x + 2y = 3\), which is equivalent to the first equation. Thus, the two equations are actually the same line. Since they share an infinite number of solutions, including the point (0,0), this system does not have a zero solution.

2. **System 2:**
- \(y = \frac{1}{2}x + 6\)
- \(-6 - 2y = x\)

Rearranging the second equation gives:
\(-2y = x + 6\) or \(y = -\frac{1}{2}x - 3\).

These two lines intersect:
- Setting \(\frac{1}{2}x + 6 = -\frac{1}{2}x - 3\).
- This leads to \(x = -3\), and substituting back gives \(y = \frac{1}{2}(-3) + 6 = 4.5\).
This has a solution but not at (0,0).

3. **System 3:**
- \(6y = -\frac{1}{2}x + 18\)
- \(8x + 4y = 2\)

Rearranging gives:
\(y = -\frac{1}{12}x + 3\) and \(y = -2x + \frac{1}{2}\).

Setting these equal for intersections does not yield (0,0) as a solution.

4. **System 4:**
- \(y = x\)
- \(y = -x\)

Setting these equations equal yields \(x = -x\), leading to \(x = 0\). Substituting gives \(y = 0\) as well. Thus, these lines intersect at (0,0), which is indeed a zero solution.

Therefore, the correct response is **System 4**: \(y = x\) and \(y = -x\), which has a zero solution.