Question

17. Modeling With Mathematics Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost $14: one small pizza, a liter of soda, and three salads cost $I5, and three small pizzas, a liter of soda, and two salads cost $22. How much does each item cost?

Answer 1

First we notice that in each order, they place the same items from the menu. Let them be

`\begin{gathered} x\rightarrow small\text{ pizza} \\ y\rightarrow liter\text{ of soda} \\ z\rightarrow salad \end{gathered}`

For the first order we know that they ordered 2 small pizzas, a liter of soda and a salad and that it cost 14 dollars. We can write this like

`2x+y+z=14`

For the second order we have one small pizza, a liter of soda and three salads. That cost 15 dollars. Then

`x+y+3z=15`

Finally, the last order was three pizzas, a liter of soda and two salads all this with a cost of 22 dollars, so

`3x+y+2z=22`

Then, from the data we have the following system of equations

`\begin{gathered} 2x+y+z=14 \\ x+y+3z=15 \\ 3x+y+2z=22 \end{gathered}`

Once we solve this system we can know how much each item cost. To solve, first we are going to solve the first equation for z. Then

`\begin{gathered} 2x+y+z=14 \\ z=14-2x-y \end{gathered}`

Then we plugg this value of z in the other two equations. Then

`\mleft\{\begin{aligned}x+y+3z=15 \\ 3x+y+2z=22\end{aligned}\rightarrow\mleft\{\begin{aligned}x+y+3(14-2x-y)=15 \\ 3x+y+2(14-2x-y)=22\end{aligned}\mright.\mright.`

Simpligying our new equations we have

`\begin{gathered} x+y+42-6x-3y=15 \\ 3x+y+28-4x-2y=22 \\ ---------------------------- \\ -5x-2y=15-42 \\ -x-y=22-28 \\ ----------------------------- \\ 5x+2y=27 \\ x+y=6 \end{gathered}`

Now we have another system of equations, but in this case we only have two equations for two variables.

To solve this new system we solve the second equation for y,

`y=6-x`

Then we plug the value of y in the first equation of our second system

`5x+2(6-x)=27`

Solving for x, we have

`\begin{gathered} 5x+2(6-x)=27 \\ 5x+12-2x=27 \\ 3x=27-12 \\ 3x=15 \\ x=(15)/(3) \\ x=5 \end{gathered}`

Now that we have tha value of x, we can find the value of y from

`\begin{gathered} y=6-x \\ y=6-5 \\ y=1 \end{gathered}`

Finally, we can know the value of z from

`\begin{gathered} z=14-2x-y \\ z=14-2(5)-1 \\ z=14-10-1 \\ z=3 \end{gathered}`

Therefore the pizza cost 5 dollars, the liter of soda 1 dollar and the salads 3 dollars.