Question

5. Caleb earns points on his credit card that he can use towards future purchases. He earns fourpoints per dollar spent on flights, two points per dollar spent at hotels, and one point perdollar spent on all other purchases. Last year, he charged a total of $9,480 and earned14,660 points. The amount of money spent on flights was $140 more than twice the amountof money spent on hotels. Find the amount of money spent on each type of purchase.can you solve it as matrix

Answer 1

This problem can be seen as the next equation

`\begin{gathered} 4F+2H+1O=T \\ F\text{ = dollars spent on flights} \\ H\text{ = dollars spent at hotels} \\ O\text{ = dollars spent in other stuff} \\ T\text{ = total of points} \end{gathered}`

Since he have charged 9480 dollars in total then we also have the next equation.

`F+H+O=9480`

With the rest of the information, we conclude the next system of equations

`\begin{gathered} 4F+2H+O=14660 \\ F+H+O=9480 \\ F-2H=140 \end{gathered}`

From the first two equations, we can substract them to obtain

`\begin{gathered} 4F+2H+O=14660 \\ - \\ F+H+O=9480 \\ = \\ 3F+H=5180 \end{gathered}`

If we multiply this last equation by 2 and add it to the last equation in the system we have

`\begin{gathered} 6F+2H=10360 \\ + \\ F-2H=140 \\ = \\ 7F\text{ = }10500 \\ \end{gathered}`

So

`\begin{gathered} F=(10500)/(7)=1500 \\ H=(1500-140)/(2)=680 \\ O=9480-1500-680=7300 \end{gathered}`

Then, he spent 1500 dlls. on flights, 680 dlls. at hotels and 7300 dlls. on others

`\begin{gathered} 4\text{ 2 1 14660 } \\ 1\text{ 1 1 9480} \\ 1\text{ -2 0 140 }\approx \\ 4\text{ 2 1 14660 } \\ 3\text{ 1 0 5180} \\ 1\text{ -2 0 140 }\approx \\ 4\text{ 2 1 14660 } \\ 3\text{ 1 0 5180} \\ 7\text{ 0 0 }10500 \\ \end{gathered}`