Solving Systems of Linear Equations using Substitution

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

Substitute

n={\color{#0000FF}{11-m}}

18({\color{#0000FF}{11-m}})+13m=178

DistrDistribute

18

198-18m+13m=178

Solve for

m

AddTermsAdd terms

198-5m=178

SubEqn

\text{LHS}-198=\text{RHS}-198

\text{-} 5m=\text{-} 20

DivEqn

\left.\text{LHS}\middle/(\text{-} 5)\right.=\left.\text{RHS}\middle/(\text{-} 5)\right.

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

Substitute

m={\color{#0000FF}{4}}

n+{\color{#0000FF}{4}}=11

SubEqn

\text{LHS}-4=\text{RHS}-4

n=7

We have found that there in the group was

4

that were members of the fan club and

7

that were not.

Solving Systems of Linear Equations using Substitution 1.11 - Solution