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# Solving Systems of Linear Equations using Substitution

## Solving Systems of Linear Equations using Substitution 1.11 - Solution

a

Let $n$ represent the number of people in the group who are not members of the fan club and $m$ represent the number of members. It is given that there are $11$ people in the group. Using this information, we can write the equation $n+m=11.$ It is also given that a non-member pays $\18$ for a ticket and a member pays $\13$ dollars, and that they together pay $\178$ for their tickets. Using this information, we can write the equation $18n+13n=178.$ Combining these equations we have the following system of equations. $\begin{cases}n+m=11 \\ 18n+13m=178 \end{cases}$

b
To find how many in the group are members of the fan club and how many are not, we have to solve the system $\begin{cases}n+m=11 & \, \text {(I)}\\ 18n+13m=178 & \text {(II)}\end{cases}$ for $n$ and $m.$ We will solve solve it using the Substitution Method. Before we can substitute, we need to isolate one of the variables in one of the equations. Let's isolate $n$ in Equation (I). $n+m=11 \quad \Leftrightarrow \quad n=11-m$ Now we can substitute $11-m$ for $n$ in Equation (II) and solve the resulting equation for $m.$
$18n+13m=178$
$18({\color{#0000FF}{11-m}})+13m=178$
$198-18m+13m=178$
Solve for $m$
$198-5m=178$
$\text{-} 5m=\text{-} 20$
$m=4$
Now that we've determined $m=4,$ we can substitute this into either of the original equations and solve the resulting equation for $n.$ For simplicity, we will use Equation (I).
$n+m=11$
$n+{\color{#0000FF}{4}}=11$
$n=7$
We have found that there in the group was $4$ that were members of the fan club and $7$ that were not.