Solving Systems of Linear Equations using Elimination

We want to solve the system $$\{\begin{array}{ll}3x-y=5& \text{(I)}\\ 8x-y=20& \text{(II)}\end{array}$$ using the elimination method. We then need to eliminate one of the variable terms by adding or subtracting the equations. We can see that the $y$-terms in this system will eliminate each other if we subtract Equation (II) from Equation (I). Let's do that! $$\begin{array}{crcl}& 3x-y& =& 5\\ -& 8x-y& =& 20\\ & \text{-}5x+0& =& \text{-}15\end{array}$$ We find $x$ by solving the resulting equation. $$\text{-}5x+0=\text{-}15\iff x=3$$ We find $y$ by substituting the value of $x$ into either of the original equations and solve the resulting equation for $y.$ Let's use Equation (I). The solution to the system of equations is $(3,4).$