Given the data from Giorgio's pizza place, we understand that there's a relationship between the quantities and costs. Essentially, we're dealing with a system of linear equations, which narrates the cost of pizzas and garlic knots:
The equations are as follows:
1) The cost of an Extra-Large pizza (x) and four garlic knots (4y) is $16
2) The cost of three Extra-Large pizzas (3x) and sixteen garlic knots (16y) is $56
Summarizing these equations:
1x + 4y = 16
3x + 16y = 56
We'll use a method known as matrix equations to solve this system. Matrix equations can be an efficient way to solve systems of equations, especially when dealing with larger quantities.
So, we'll convert these equations into a matrix of the form `AX=B`, where `A` is the matrix of coefficient, `X` is the column matrix of variables (x, y) and `B` is the column matrix of constants (16, 56).
Our `A`, `X` and `B` matrices are:
|1, 4 | |x| | 16 |
|3, 16| * |y| = | 56 |
Now, to solve this matrix equation for `X`, we'll calculate the inverse of `A` (denoted as `A^-1`) and then multiply it by `B`. This results `X = A^-1 * B`
By solving this calculation, here's what we get:
The cost of an Extra-Large pizza (x) is $8.0 and the cost per garlic knot (y) is $2.0.
Hence, each Extra-Large pizza costs $8 and each garlic knot costs $2 at Giorgio's pizza place.