Elimination method review (systems of linear equations)

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.

We're asked to solve this system of equations:

We notice that the first equation has a $7x$7, x term and the second equation has a $-7x$minus, 7, x term. These terms will cancel if we add the equations together—that is, we'll eliminate the $x$x terms:

Solving for $y$y, we get:

Plugging this value back into our first equation, we solve for the other variable:

The solution to the system is $x=\blueD{-1}$x, equals, start color #11accd, minus, 1, end color #11accd, $y=\goldD{1}$y, equals, start color #e07d10, 1, end color #e07d10.

We can check our solution by plugging these values back into the original equations. Let's try the second equation:

Yes, the solution checks out.

We're asked to solve this system of equations:

We can multiply the first equation by $-4$minus, 4 to get an equivalent equation that has a $\purpleD{-16x}$start color #7854ab, minus, 16, x, end color #7854ab term. Our new (but equivalent!) system of equations looks like this:

Adding the equations to eliminate the $x$x terms, we get:

Solving for $y$y, we get:

Plugging this value back into our first equation, we solve for the other variable:

The solution to the system is $x=\blueD{5}$x, equals, start color #11accd, 5, end color #11accd, $y=\goldD{0}$y, equals, start color #e07d10, 0, end color #e07d10.