The Elimination Method is an algebraic process of multiplying each equation by suitable constants in order to have the same \(x\)-terms, or \(y\)-terms, but with opposite signs. This allows the two equations to be added together and these terms will cancel out. The single equation generated will have only one variable that can be solved for.
Solve the system \(\begin{array}{l} -6x+2y=22 \\ 2x+8y=36 \end{array}\) using the Elimination Method.
The first equation has \(-6x\) and the second equation has \(2x\). Since these terms have opposite signs already, that will be the variable to try and eliminate.
| Multiply equation two by \(3\): | \(3\cdot\left(2x+8y=36\right)\implies 6x+24y=108\) |
| Now add the resulting equations to eliminate \(x\): | \(\begin{array}{l} -6x+2y=22 \\ \underline{+6x+24y=108} \\ \hfill 26y=130 \end{array}\) |
| Solve the new equation for \(y\): | \(\large{\frac{26y}{26}}\normalsize{=}\large{\frac{130}{26}}\normalsize{\implies y=5}\) |
Now that the \(y\)-coordinate of the solution has been found, substitute the value back into ANY original equation and solve for \(x\):
| Use the first equation: | \(\begin{array}{l} -6x+2y=22 \\ \implies -6x+2(5)=22 \\ \implies -6x+10=22 \\ \implies -6x=12 \\ \implies \large{\frac{-6x}{-6}}\normalsize{=}\large{\frac{12}{-6}} \\ \implies x=-2 \end{array}\) |
| Hence the solution to the system is the point \((-2,5)\;\color{green}{\checkmark}\). | |