Answer:
Explanation:
You want to know the numbers of small ($6), medium ($10), and large ($11) pizzas sold, if 52 pizzas were sold for $419, and 7 more large than medium were sold.
Equations
The system of equations describing the scenario can be written as ...
- x + y + z = 52
- y -z = -7
- 6x +10y +11z = 419
Elimination
Gaussian elimination systematically reduces these equations so that each has one less variable than the one before. Here, that is already true of the first two equations. We can use the first two equations to eliminate the x- and y-variables in the third equation. The operation we will use is ...
- Subtract 6 times "row 1" and subtract 4 times "row 2" from "row 3" to get the reduced third equation.
- (6x +10y +11z) -6(x +y +z) -4(y -z) = 419 -6(52) -4(-7)
- 9z = 135
This equation can be solved for z to get ...
z = 135/9 = 15
Back substitution
Using this value of z in "row 2", we can find y:
y -(15) = -7
y = 8 . . . . . . . add 15
Using the values of y and z in "row 1", we can find x:
x +8 +15 = 52
x = 29
The pizzeria sold 29 small, 8 medium, and 15 large pizzas.
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Additional comment
In matrix terms, the Gaussian elimination method starts with the augmented matrix for the system of equations, then reduces it to "row-echelon form", an upper triangular form. The diagonal elements are all made to be equal to 1. The attachment shows the result of a calculator doing this.
The usual algorithm starts by swapping rows to put the largest column 1 value in the first row. Then the row is normalized by dividing it by that value. For this problem, the largest x-coefficient is 6, so the calculator swapped rows 1 and 3 to start, then divided the new row 1 by 6.
In our manual solution, above, we avoided fractions by choosing diagonal elements that were already 1.
The purpose of starting by putting the largest number on the diagonal is to reduce numerical errors that can occur during the computation by a calculator of computer. When exact rational arithmetic is used, that is not an issue.
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