Elimination Method

Method

Two equations - two unkown variables

The central idea of this method is to add or subtract one equation from the other in such a way that one of the variables is eliminated. Then, the remaining equation with only one variable left is used to find the value of this variable.

Once, one variable is known it can be substituted into any one of the original equations. Which can now be solved for the second, remaining variable.

example 1

$${\begin{align} x + y & = 20 \newline x - y & = 2 \end{align}}$$

Adding the LHS and RHS together

$${\begin{align} x + y + (x - y) & = 20 + 2 \end{align}}$$

$${\begin{align} 2x & = 22 \newline x & = 11 \end{align}}$$

example 2

$${\begin{align} 4x + 2y & = 20 \newline x - y & = 2 \end{align}}$$

If we add (or subtract) one equation from the other what do we get?

Do you get

$${\begin{align} 5x + y & = 22 \end{align}}$$

Have you eliminated one of the variables? No.

Try multiplying the second equation by 2.

$${\begin{align} 4x + 2y & = 20 \newline \color{green}{2}x - \color{green}{2}y & = \color{green}{2} \cdot 2 \end{align}}$$

$${\begin{align} 4x + 2y & = 20 \newline \color{green}{2}x - \color{green}{2}y & = 4 \end{align}}$$

Now, adding will eliminate the variable $y$.

$${\begin{align} 6x & = 24 \end{align}}$$

$${\begin{align} x & = 4 \end{align}}$$

So, substituting $\color{green}{x = 4}$ into one of the equations.

$${\begin{align} 4\color{green}{\cdot 4} + 2y & = 20 \newline 16 + 2y & = 20 \newline 2y & = 4 \newline y & = 2 \end{align}}$$

example 3

TODO

Unit 5 Section 5 : Simultaneous Equations


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