Question

Which system of equations below has exactly one solution?
y = –8x – 6 and y = –8x + 6
y = –8x – 6 and One-halfy = –4x – 3
y = –8x – 6 and y = 8x – 6
y = –8x – 6 and –y = 8x + 6

Answer 1

To determine which system of equations has exactly one solution, we need to analyze the relationships between the equations in each pair.

1. **\(y = -8x - 6\) and \(y = -8x + 6\)**:
These two equations have the same slope (-8), which means they are parallel lines. Since parallel lines never intersect, this system has no solutions.

2. **\(y = -8x - 6\) and \( \frac{1}{2}y = -4x - 3\)**:
First, we can rewrite the second equation in terms of y:
\[
\frac{1}{2}y = -4x - 3 \implies y = -8x - 6
\]
This means both equations are actually the same line. Hence, there are infinitely many solutions.

3. **\(y = -8x - 6\) and \(y = 8x - 6\)**:
In this case, the slopes are -8 and 8, so these lines are not parallel and will intersect at exactly one point. This system has exactly one solution.

4. **\(y = -8x - 6\) and \(-y = 8x + 6\)**:
Rewriting the second equation gives:
\[
y = -8x - 6
\]
Again, both equations are the same line, leading to infinitely many solutions.

From the analysis, the system that has exactly one solution is:

**\(y = -8x - 6\) and \(y = 8x - 6\)**.