Elimination vs. Substitution

When solving systems of equations, we want to find the simplest method possible. Depending on the equations being solved for, different methods help. Here are two general rules to help determine the best possible system of solving.

1. If in either equation, one variable has already been solved for, use the substitution method.

-This is because when one variable is already solved for, it is much simpler to plug in the value of x or y and from there, solve the equation.

For example:

y = 3 x - 5

y - 3 = 3 (x - 7)

In this case, Y has been solved for. Simply plug in or "substitute" the value of y (3x-5) in for the Y of the second equation.

2. If the X values or Y values of both equations cancel out, or if the values can be multiplied simply so they do cancel out, use elimination.

-This is because if the X's or Y's simply cancel out, the equations will be much easier to solve for because only one variable is left to deal with.

For example:

5x + 4y = 20

3x - 4y = 60

Notice the Y variables cancel out because one is a positive 4 while the other is a negative.

Also:

3x + y = 10

-9x + 3y = 30

By observing carefully, you can see that one of the equations can be multiplied so that one of the variable will fall out. Which equation is it? What can it be multiplied by?

Multiple everything in equation one by 3. By doing this, the coefficient to X becomes positive 9, which is directly lined up with a negative 9 causing them to cancel out.

Using these two rules, solving systems of equations will turn out to be much simpler.