9Example: substitution method

Now look at the equations that give the ages of Tina's father and mother.

Solve the problem using the substitution method:

$\def\arraystretch{1.25} \begin{array}{lcccl}\mathrm{I} &2v&+&1.5m&=&150&\\\mathrm{II} &m&+&3&=&v-2\end{array}$

Step 1: Resolve equation $\mathrm{II}$

Which equation you solve for which unknown is partly a "sharp look", partly a matter of taste! Here the equation $\mathrm{II}$ is solved for $v$.

$\def\arraystretch{1.25} \begin{array}{lrcll}\mathrm{II}& m+3&=&v-2& \qquad|+2\\\mathrm{II}'& v&=&\color{#009900}{m+5}&\end{array}$

Step 2: Plug $\mathrm{II}'$ into $\mathrm{I}$

$\def\arraystretch{1.25} \begin{array}{rrll}\mathrm{I}& 2\color{#009900}{(m+5)} + 1.5m &= &150\\\mathrm{I}& 2m+10 + 1.5m &= &150\\\mathrm{I}& 3.5m+10&= &150&\qquad |-10\\\mathrm{I}&3.5m&=&140& \qquad|:3.5\\\mathrm{I}&\color{#cc0000}m&=&\color{#cc0000}{40}\end{array}$

Step 3: Plug m into $\mathrm{II}'$

$\def\arraystretch{1.25} \begin{array}{rrl}\mathrm{II}'&v&=&\color{#cc0000}{40} + 5\\\mathrm{II}'&\color{#009999}{v}&=&\color{#009999}{45}\end{array}$

Tina now knows how old her parents are thanks to you. Her mother is $\color{#cc0000}{40}$ years old and her father is $\color{#009999}{45}$ years old.

You can also write this down formally as a solution set:

$\mathbb{L}=\{(m|v)=(\color{#cc0000}{40}|\color{#009999}{45})\}$.