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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.
a
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

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:

[tex]\left[\begin{array}{ccc|c}1&1&1&45\\1&-2&0&0\\50&120&40&2800\end{array}\right][/tex]

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.

[tex]\left[\begin{array}{ccc|c}1&1&1&45\\0&-3&-1&-45\\0&70&-10&550\end{array}\right][/tex]

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.

[tex]\left[\begin{array}{ccc|c}1&1&1&45\\0&-3&-1&-45\\0&0&-\dfrac{100}{3}&-500\end{array}\right][/tex]

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

[tex]\left[\begin{array}{ccc|c}1&1&1&45\\0&-3&0&-30\\0&0&1&15\end{array}\right][/tex]

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

[tex]\left[\begin{array}{ccc|c}1&0&0&20\\0&1&0&10\\0&0&1&15\end{array}\right][/tex]

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