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.