Substitution Method

Method

Make one of the variables in one equation (say I) the subject, i.e., solve for one variable. Then, substitute the expression on the other side for that variable in the second equation (II)

example 1

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

Rearrange one of the equations making one variable the subject.

$${\begin{align} y & = \color{green}{20 - x} ~ (I) \newline x - y & = 2 ~ ~ ~(II)\end{align}}$$

Substitute $\color{green}{20 - x}$ for y in the second equation:

$${\begin{align} x - (\color{green}{20 - x}) & = 2 ~ ~ ~(II)\end{align}}$$

Solve for the variable that is left.

$${\begin{align} 2x - 20 & = 2 ~ ~ ~(II) \newline 2x & = 22 \newline x & = 11\end{align}}$$

Finally, substitute $\color{green}{x= 11}$ into one of the original equations to solve for y (the remaining unknown variable).

$${\begin{align} \color{green}{11} + y & = 20 ~ (I) \newline y & = 9 \end{align}}$$

Solution is $ x = 11, y = 9 $

Lösungsmenge $\mathbb{L} = \left\{ x = 11, y = 9 \right\} $


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