Linear systems are groups of equations with x and y variables. The variables correspond to the x-axis and y-axis on a graph. Calling the equations "linear" simply means the results can be graphed from the solved x and y coordinates. The elimination method is one of many ways to solve linear systems. The basic explanation is that you add the equations together to eliminate one variable.
Write down your equations.
Solve both equations so they are in the form y = equation.
Look at both equations carefully to see if either the x or y variable can be eliminated if the two equations were added together.
Add the two equations together to create one new equation with only one variable.
Solve the equation from Step 4.
Substitute the result from Step 5 into one of the original equations to solve for your x variable.