Look for any $x$-values of $0$ to find some sort of constant. In this case, an $x$-value of $0$ equates to a $y$-value of $-3$, so we can include the $-3$ value in the rule.
Since subtracting $3$ does not give the correct $y$-value for the rest of the table, there are more parts to this rule.

We can now try working backwards to find the other missing parts. If the $y$-value was obtained by subtracting $3$ among other things, we can add $3$ to the $y$-value and see what other changes took place as well.
Using the $x$-value $−2$ and y-value $−7$:
Adding $3$ to $−7 = −4$
$−4$ is two times $−2$, the $x$ -value.

By adding $2x$ to the rule, we get $y = 2x − 3$.

Test to see if this rule is correct by substituting in $7$ for the $x$-value and see if you get $11$ as the $y$-value. If it is, use the rule to fill in the rest of the table. One value has been given to you (when $x = 4$, $y = 5$).