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