A baker must produce three types of cakes for different events he has been hired for this week. He charges $50 for birthday cakes, $120 for wedding cakes, and $40 for an assorted batch of cupcakes.
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The baker must fulfill 45 orders and will earn a total of $2,800. The baker has to bake twice as many birthday cakes as wedding cakes. Write a system of equations and use a matrix to find the total number orders for each type of cake

Step-by-step explanation:

Let the variables b, w, c stand for the numbers of birthday cakes, wedding cakes, and cupcake batches, respectively. We can write equations based on the described relations.

  b + w + c = 45 . . . . . . the baker must fill 45 orders

  b -2w   =   0 . . . . . . .  birthday cakes are 2 times wedding cakes

  50b +120w +40c = 2800 . . . . revenue from sales is 2800

These equations can be put in the form of an augmented matrix:

We can begin the row-reduction process by subtracting the first row from the second, and by subtracting 50 times the first row from the third.

The next step is to multiply the second row by 70/3 and add that to the third row. Now we have an upper triangular matrix.

We can multiply the third row by -3/100, then add that result to the second row. This gives ...

Dividing the second row by -3, then subtracting the second and third rows from the first completes the solution.

The baker has orders for 20 birthday cakes, 10 wedding cakes, and 15 batches of cupcakes.