If $(x,y)$ is the solution to the following system of equations, what is the value of $10x-y$? $$5x+3y=31$$ $$5x-4y=17$$

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If we compare the given equations to the requested expression, we see a clear relationship.

$$5x+3y=31$$ $$5x-4y=17$$ $$\begin{array}{ c c c} &5x & 3y & 31 \\ +&5x & -4y & 17 \\ \hline \end{array}$$ $$10x-y=\boxed{48}$$

We can use elimination to find the values of $x$ and $y$.

$$5x+3y=31$$

Multiply each term of the second equation by $-1$.

$$-1(5x-4y)=17(-1)$$ $$-5x+4y=-17$$

Add the two equations.

$$\begin{array}{ c c c} &5x & 3y & 31 \\ +&-5x & 4y & -17 \\ \hline \end{array}$$ $$7y=14$$ $$y=2$$

Substitute the value of $y$ back into either equation.

$$5x+3(2)=31$$ $$5x+6=31$$ $$5x=25$$ $$x=5$$

Using the values of $x$ and $y$,

$$10x-y$$ $$=10(5)-2$$ $$=50-2$$ $$=\boxed{48}$$

Questions like these are especially common on standardized tests like the SAT. A good hint that you would use the first approach would be if the question requests the value of an expression, rather than the $x$ or $y$ coordinate.