Question

$y = \frac{1}{2}x - 1$
$y = 5 - x$
how many solutions does this system of equations have?
infinitely many solutions, because the slopes are the same and the $y$-intercepts are the same.
no solution, because the slopes are the same and the $y$-intercepts are different.
one solution, because the slopes are different.

Explanation:

Step1: Identify slopes of equations

For $y=\frac{1}{2}x - 1$, slope $m_1=\frac{1}{2}$. For $y=5 - x$, slope $m_2=-1$.

Step2: Compare slopes

Since $\frac{1}{2}
eq -1$, the slopes are different.

Step3: Relate slopes to solutions

Linear equations with different slopes intersect at exactly one point, so there is one solution.

Answer:

One solution, because the slopes are different.