Question

Use Gaussian elimination to solve.The burkes pay their babysitter $5 per hour before 11p.m. and $7.50 after 11 p.m. One evening they went out for 5 hours and paid the sitter $35.00. Whag time did they come home?

Answer 1

Given the information on the problem, we can write the following system of equations:

$\begin{cases}x+y=5 \\ 5x+7.5y=35\end{cases}$

then, we can write the following augmented matrix:

$\begin{bmatrix}{1} & {1} & {5} \\ {5} & {7.5} & {35}\end{bmatrix}$

now, if we multiply by 5 the first row and then substract it from the second row, we get:

$\begin{bmatrix}{1} & 1{} & {5} \\ {5} & {}7.5 & 35{}{}\end{bmatrix}\rightarrow\begin{bmatrix}{1} & 1{} & 5{} \\ {0} & {2.5} & {10}{}\end{bmatrix}$

notice that from the second equation, we can find the value of y:

$\begin{gathered} 2.5y=10 \\ \Rightarrow y=\frac{10}{2.5}=4 \\ y=4 \end{gathered}$

now that we have that y = 4, we can use this value on the first equation to find x:

$\begin{gathered} x+4=5 \\ \Rightarrow x=5-4=1 \\ x=1 \end{gathered}$

now, we have that the babysitter worked 1 hour before 11pm and 4 hours after 11 pm,then, the Burkes came home at 3 am