Question

Which of the following systems describes a zero solution
2y + x =3 and 4x + 7y =12
y=x and y =-x
6y=-12x +18 and 8x+4 y= 2
y =1/2x+6 and-6-2y=x

Answer 1

To find which of the given systems describes a zero solution (where \(x=0\) and \(y=0\)), we need to substitute \(x=0\) and \(y=0\) into each system of equations and check if they hold true.

1. **System 1:**
\[
2y + x = 3 \\
4x + 7y = 12
\]
Substituting \(x=0\) and \(y=0\):
\[
2(0) + 0 = 3 \quad \Rightarrow \quad 0 \neq 3 \\
4(0) + 7(0) = 12 \quad \Rightarrow \quad 0 \neq 12
\]
**Not a zero solution.**

2. **System 2:**
\[
y = x \\
y = -x
\]
Substituting \(x=0\) and \(y=0\):
\[
0 = 0 \quad \text{(True)} \\
0 = -0 \quad \text{(True)}
\]
**This is a zero solution.**

3. **System 3:**
\[
6y = -12x + 18 \\
8x + 4y = 2
\]
Substituting \(x=0\) and \(y=0\):
\[
6(0) = -12(0) + 18 \quad \Rightarrow \quad 0 = 18 \quad \Rightarrow \quad 0 \neq 18 \\
8(0) + 4(0) = 2 \quad \Rightarrow \quad 0 \neq 2
\]
**Not a zero solution.**

4. **System 4:**
\[
y = \frac{1}{2}x + 6 \\
-6 - 2y = x
\]
Substituting \(x=0\) and \(y=0\):
\[
0 = \frac{1}{2}(0) + 6 \quad \Rightarrow \quad 0 \neq 6 \\
-6 - 2(0) = 0 \quad \Rightarrow \quad -6 \neq 0
\]
**Not a zero solution.**

In conclusion, **the only system that describes a zero solution (where \(x=0\) and \(y=0\)) is:**
- **System 2: \(y = x\) and \(y = -x\)**.