Example What is the solution of the system of equations? \[ \begin{array}{l} -5 x+2 y=22 \\ 10 x+2 y=-8 \end{array} \] \[ 2(-5 x+2 y=22) \rightarrow \begin{array}{rlrl} -10 x+4 y & =44 & 10 x+2 y & =-8 \\ +10 x+2 y & =-8 \\ \hline 6 y & =36 & 10 x+2(6) & =-8 \\ y & =6 & 10 x+12 & =-8 \\ & 10 x & =-20 \\ x & =-2 \end{array} \] The solution is \( (-2,6) \). (1) Show a different way to use elimination to solve the system of equations in the Example.

Let’s switch it up a bit and tackle the problem with a fresh approach! Start with the two equations: \[ -5x + 2y = 22 \quad (1) \] \[ 10x + 2y = -8 \quad (2) \] To eliminate \(y\), let's subtract equation (1) from equation (2): \[ (10x + 2y) - (-5x + 2y) = -8 - 22 \] This simplifies to: \[ 15x = -30 \] Dividing both sides by 15 gives us: \[ x = -2 \] Now plug \(x = -2\) back into either equation to solve for \(y\). Using (1): \[ -5(-2) + 2y = 22 \implies 10 + 2y = 22 \implies 2y = 12 \implies y = 6 \] So again, we find the solution is \( (-2, 6) \)! Remember, there are often multiple methods to arrive at the same solution—feel free to mix and match techniques, like substitution or graphing, to keep things interesting!