Question

At a vegetable stand, a person can buy 4 apples and 2 bags of potatoes for $9. At a farmer’s market, the same person can buy 4 apples and 3 bags of potatoes for $12. Assuming the prices are the same, what is the price of each apple and each bag of potatoes? Use the matrix equation to solve the system of equations.

Answer 1

y = 3 and x =
`(5)/(4)`

Explanation:

1. First lets construct two simultaneous equations

4x + 2y = 9 (x for apples and y for potatoes) and

4x + 3y = 12 (even now its pretty obvious to see that a potato costs 3 dollars however pretend you dont know that and lets use the matrix equation)

2. now lets put everything into matrices

`\left[\begin{array}{ccc}4&2\\4&3\\\end{array}\right]`
`\left[\begin{array}{ccc}x\\y\end{array}\right]` =
`\left[\begin{array}{ccc}9\\12\end{array}\right]`

The equation states that

`\left[\begin{array}{ccc}x\\y\end{array}\right]` =
`A^(-1)` .
`\left[\begin{array}{ccc}9\\12\end{array}\right]`

where A =
`\left[\begin{array}{ccc}4&2\\4&3\\\end{array}\right]`

3. To find the inverse of A we first need to find the determiner of A and to find the determiner of A we need to do

4 x 3 - 4 x 2 =

4 therefore we will be multiplying our inverse matrix by
`(1)/(4)`

4. Now we know that
`A^(-1)` =
`(1)/(4)` x
`\left[\begin{array}{ccc}3&-2\\-4&4\\\end{array}\right]`

5. Lastly we have to use our equation

`\left[\begin{array}{ccc}x\\y\end{array}\right]` =
`(1)/(4)` x
`\left[\begin{array}{ccc}3&-2\\-4&4\\\end{array}\right]`
`\left[\begin{array}{ccc}9\\12\end{array}\right]`

so the values are as follows

`(1)/(4)` (3 x 9 + -2 x 12) =
`(5)/(4)`

so x =
`(5)/(4)`

and to find y we multiply the bottom part of the matrices

`(1)/(4)` (-4 x 9 + 4 x 12) =
`(12)/(4)` or 3

so y = 3 (which we confirmed at the start)