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

One alternative approach using elimination is to align the equations by eliminating one variable directly. Start with the original equations: 1. \(-5x + 2y = 22\) 2. \(10x + 2y = -8\) To eliminate \(y\), subtract the first equation from the second. This will give you: \[ (10x + 2y) - (-5x + 2y) = -8 - 22 \] This simplifies to: \[ 15x = -30 \implies x = -2 \] Now substitute \(x = -2\) back into one of the original equations. Using the first one: \[ -5(-2) + 2y = 22 \implies 10 + 2y = 22 \implies 2y = 12 \implies y = 6 \] Thus, the solution remains \( (-2, 6) \). A fun tip with elimination is to keep an eye on coefficients; if they're out of sync, multiplying one or both equations to match them can create a clean path to simplification. Remember, avoidance of mistakes in sign handling and arithmetic is key—double-check your calculations!