Question

Which of the following systems describes a zero-solution?

Answer Options:

y = (1/2)x + 6 and -6 - 2y = x
2y + x = 3 and 4x + 8y = 12
6y = -12x + 18 and 8x + 4y = 2
y = x and y = -x

Answer 1

To identify which of the given systems has a zero-solution (meaning the system has no solutions), we need to analyze each pair of equations to see if they can be satisfied simultaneously.

1. **System 1:**
\[
y = \frac{1}{2}x + 6
\]
\[
-6 - 2y = x \quad \text{(or, rearranging: } x = -6 - 2y)
\]

Substitute \(y\) from the first equation into the second:
\[
x = -6 - 2\left(\frac{1}{2}x + 6\right)
\]
\[
x = -6 - (x + 12)
\]
\[
x = -6 - x - 12
\]
\[
2x = -18 \quad \Rightarrow \quad x = -9
\]
Substitute \(x = -9\) back into the first equation to find \(y\):
\[
y = \frac{1}{2}(-9) + 6 = -\frac{9}{2} + 6 = \frac{3}{2}
\]
Thus, this system has one solution: \((-9, \frac{3}{2})\).

2. **System 2:**
\[
2y + x = 3 \quad \text{(or, rearranging: } x = 3 - 2y)
\]
\[
4x + 8y = 12 \quad \text{(This can be simplified by dividing by 4: } x + 2y = 3)
\]

Both equations are actually the same \(x + 2y = 3\). Therefore, this system has infinitely many solutions.

3. **System 3:**
\[
6y = -12x + 18 \quad \text{(or, rearranging: } y = -2x + 3)
\]
\[
8x + 4y = 2 \quad \text{(This can be simplified: } 2x + y = \frac{1}{2})
\]

Substitute \(y\) from the first equation into the second:
\[
2x + (-2x + 3) = \frac{1}{2}
\]
\[
3 = \frac{1}{2}
\]
This is a contradiction, meaning this system has no solutions.

4. **System 4:**
\[
y = x
\]
\[
y = -x
\]

These two lines intersect at the origin \((0, 0)\), so there is one solution.

From the analysis above, **System 3** \(6y = -12x + 18\) and \(8x + 4y = 2\) describes a zero-solution, as it has no solutions (a contradiction arises).

Thus, the answer is **System 3**.