Methods of Solving Systems of Equations with 2 Unknowns

Elimination Method

In the elimination method (also known as the addition-subtraction method), we eliminate one of the unknowns by first (if necessary) multiplying each equation by a factor will make the coefficients of one unknown in both equations equal, regardless of their signs. The two equations are then added or subtracted so as to eliminate that variable.

Let’s watch a video on how it’s done:

  • Note that the arithmetic error made at time 3:21 – bringing 1 over to the other side makes it 3 minus 1 = 2.

Sometimes you might get coefficients for x and y that are fractions, as is the case in the video below. When this occurs, you can still perform the elimination method as you did above, expect it’s best to first rewrite the whole equation by finding a lowest common denominator among the terms. This is demonstrated below:

SUMMARY – To solve a linear system by elimination, follow these steps:

1)   Arrange the two equations so that like terms are aligned.

2)   Choose the variable you wish to eliminate.

  • If necessary, multiply one or both equations by a value so that they have the same or opposite coefficient in front of the variable you want to eliminate.

3)   Add or subtract (as needed) to eliminate one variable.

4)   Solve for the remaining variable.

5)   Substitute into one of the original equations to find the value of the other variable.

6)   Check your solution by substituting into the original equations, or into the word problem.

7)   If you are solving a word problem, write the answer in words.

Substitution Method

This next method, called substitution, involves solving a linear system by substituting for one variable from one equation into the other equation. Remember that with each new method, you have more options for solving the linear system. You could, by all means, use elimination to solve every linear system you come across (or substitution), so it’s up to you to choose which method is best for which situation.

When it comes to deciding which method is best, it’s wiser to choose substitution over elimination if you’re given an equation where a variable is already isolated for. Take for example:

x + 3y = 5    (1)

y = 4x – 1   (2)

Notice that in equation (2), y is already isolated. Therefore, you’re better-off using substitution since you won’t have to shift the terms around until they’re aligned, as you would if you were eliminating.

The final video below shows how to tackle linear systems that have fractions as their coefficients. The best way to approach these questions is to multiple the whole equation containing the fraction by the lowest common denominator among the terms. This will make the equation easier to work with because this initial technique converts the equation into integer coefficients.