Question

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

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

6y=−12x+18 and 8x+4y=2
6 y equals negative 12 x plus 18 and 8 x plus 4 y equals 2

y=x and y=−x
y equals x and y equals negative x

y=12x+6 and −6−2y=x

Answer 1

To determine which system of equations describes a zero-solution (meaning that the system has no solution), we will analyze each system for inconsistency.

1. **System 1**:
\[
2y + x = 3
\]
\[
4x + 8y = 12
\]
The second equation can be simplified by dividing everything by 4:
\[
x + 2y = 3
\]
This is equivalent to the first equation, which means these two lines are the same. Therefore, there are infinitely many solutions (not zero).

---

2. **System 2**:
\[
6y = -12x + 18 \quad(1)
\]
\[
8x + 4y = 2 \quad(2)
\]
Rewriting the first equation gives:
\[
y = -2x + 3
\]
For the second equation, solving for y:
\[
4y = 2 - 8x \implies y = \frac{1}{2} - 2x
\]
Now we set the two expressions for y equal to find any possible solutions:
\[
-2x + 3 = \frac{1}{2} - 2x
\]
After simplifying, both equations lead to a false statement (such as 3 = 0), indicating that this system has no solutions (zero-solution).

---

3. **System 3**:
\[
y = x
\]
\[
y = -x
\]
These two equations are lines that intersect at the origin (0,0). Therefore, there is one solution (not zero).

---

4. **System 4**:
\[
y = 12x + 6
\]
\[
-6 - 2y = x \implies 2y = -x - 6 \implies y = -\frac{1}{2}x - 3
\]
These two lines have different slopes (one has a slope of 12, the other has a slope of -1/2) and will intersect at some point. Thus, there is one solution (not zero).

---

After analyzing all the systems, **System 2** (6y = -12x + 18 and 8x + 4y = 2) is the only one that yields a zero-solution (no solution).