Question

how many solutions does the system of equations below have?
$6x + 2y = -5$
$15x + 5y = -8$
options:
no solution
one solution
infinitely many solutions

Explanation:

Step1: Rewrite equations in slope-intercept form

For \(6x + 2y=-5\), solve for \(y\):
\(2y=-6x - 5\)
\(y=-3x-\frac{5}{2}\)

For \(15x + 5y=-8\), solve for \(y\):
\(5y=-15x - 8\)
\(y=-3x-\frac{8}{5}\)

Step2: Analyze slopes and intercepts

Both lines have a slope of \(-3\) (equal slopes), but their \(y\)-intercepts are \(-\frac{5}{2}\) and \(-\frac{8}{5}\) (unequal). Parallel lines (same slope, different intercepts) never intersect, so the system has no solution.

Answer:

no solution