Invent a system given its solution (elimination)

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Invent a system of two equations whose solution is $x=-3$ and $y=7$, containing terms in both $x$ and $y$ in each equation. Then solve it using the elimination method.

Solution

Build two equations by requiring the pair $(-3;7)$ to satisfy them. For instance, starting from $x+y$ and $2x-y$: $x+y=-3+7=4, \qquad 2x-y=2(-3)-7=-13.$ A valid system is therefore $\begin{cases} x+y=4\\ 2x-y=-13 \end{cases}$ Check by elimination. Adding the two equations term by term eliminates $y$: $3x=-9 \implies x=-3$. Substituting into the first: $-3+y=4 \implies y=7$. $\boxed{\begin{cases} x+y=4\\ 2x-y=-13 \end{cases}\ \Rightarrow\ x=-3,\ y=7}$ (Any other pair of independent equations satisfied by $(-3;7)$ is equally correct.)