The table below represents a linear relationship between \(x\) and \(y\), where \(a\) is a constant. Which equation represents the relationship in the table?

$$ \begin{array}{|c||c|c|c|} \hline x & a & 3a & 5a \\ \hline y & 0 & -a & -2a \\ \hline \end{array} $$

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As with other table of values, we can choose two points and find the slope between them. Using the points \((a,0)\) and \((3a,-a)\),

$$ m=\frac{-a-0}{3a-a} $$ $$ m=\frac{-a}{2a} $$ $$ m=-\frac{1}{2} $$

We can use either the point-slope equation or the slope-intercept equation to solve, as was shown in problem 1. Since the answer choices are in slope-intercept form, lets use that method this time.

$$ y=mx+b $$

Substituting in our values \(m=-\frac{1}{2}\) and \((a,0)\), we get that

$$0=-\frac{1}{2}(a)+b$$ $$\frac{a}{2}=b $$

Substituting \(m\) and \(b\) into the slope-intercept equation results in the correct answer.

We can always plug in the given \((x,y)\) coordinates into the four equations. We can quickly tell that the first two choices do not work for \((a,0)\).

\((3a,-a)\) only works for the third choice.