Applications of linear systems

Getting equations from the problem setting

We have:

• 30 children

• 4 times as many girls as boys

Set the variables.

$m:$ number of girls

$j:$ number of boys

There are 30 children in total. Each child is either a girl or a boy.

$\mathrm{I}m+j=30$

The number of girls is four times the number of boys.

Careful: You have to multiply the number of boys by 4, not the number of girls!

$\mathrm{II}4j=m$

You now have a system of equations with two equations and two variables.

Recall that there are three possible solution methods.

In this solution, we will use the substitution method is used.

Substitution method

Step 1: Solve for one variable

Look at the second equation, this one is already solved for $m$.

Step 2: Plug $m$ into $\mathrm{I}$.

$m$ in $\mathrm{I}$

$\begin{array}{rrl}4j+j& =& 30\\ 5j& =& 30\\ j& =& 6\end{array}$

Step 3: Plug $j$ into $\mathrm{II}$.

$j$ in $\mathrm{II}$

$\begin{array}{rrl}4\cdot 6& =& m\\ 24& =& m\end{array}$

Write down the solution set.

$𝕃=\{(j|m)=(6|24)\}$

So there are $6$ boys and $24$ girls on the party.