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Solve Systems of Equations with the Elimination Method

The elimination method demands a bit more understanding than the substitution method. The disadvantage is that it can become difficult if you don’t have “nice” numbers to work with. The number of columns you use during the calculations can be used as a convenient way for you to separate the two methods.

Rule

Recipe for the Elimination Method

1.: Choose which of the variables you want to get rid of.
2.: Multiply each of the equations (1) and (2) together with the numbers that give the variable you chose the same constant in both equations, but with opposite signs.
3.: Write the new forms of the equations below the original ones.
4.: Add equation (2) to equation (1) and write the result below those equations. You now have one equation with one variable. Solve it.
5.: Put the answer you found back into equation (1) and solve for the last variable.
6.: Write the answers on coordinate form:
$ANSWER: (x, y) = (a, b)$

Example 1

Solve the system of equations.

\begin{align} y + 2 x & = 1 & (1) \\ 2 y - x & = 2 & (2) \end{align}

1.

We decide to get rid of

y

2.

Because you have

2 y

in (2) and

y

in (1), you have to multiply (1) by

- 2

to get rid of the

y

value when you add the two equations together. In this case, you don’t need to multiply (2) by any factors.

\begin{align} y + 2 x & = 1 | \cdot (- 2) & (1) \\ 2 y - 4 x & = 2 & (2) \end{align}

3.

Write the two equations again after making the changes.

\begin{array}{l} - 2 y - 4 x & = - 2 \\ 2 y - x & = 2 \end{array}

4.

Now, add the two equations together to get rid of

y

, and solve for

x

\begin{array}{l} - 2 y + 2 y - 4 x - x & = - 2 + 2 \\ 0 y - 5 x & = 0 \\ x & = 0 \end{array}

5.

Insert this answer into one of the equations. You can choose whichever one you like. I chose equation (1):

\begin{array}{l} y + 2 x & = 1 \\ y + 2 \cdot (0) & = 1 \\ y + 0 & = 1 \\ y & = 1 \end{array}

6.

Write the answer on coordinate form:

(x, y) = (0, 1)

Solve Systems of Equations with the Substitution Method

How to Solve Systems of Nonlinear Equations

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