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

1 answer

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 \(y = 8x - 6\).