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Apples cost $a per kilo, bananas cost $b per kilo, and coconuts cost $c each.

Annie spends a total of $26 on 2 kilos of apples, 2 kilos of bananas, and 1 coconut. Bob spends a total of $40 on 4 kilos of apples, no bananas, and 2 coconuts. Carol spends a total of $18 on 3 kilos of apples, 1 kilo of bananas, and no coconuts.

(a) What is an equation that defines Annie's total spending as a function of $a$, $b$, and $c$? What is a similar equation for Bob? Carol?

(b) Write down the system of equations from part (a) in matrix form.

(c) Solve for $a$, $b$, and $c$ by using the adjoint method.

Answer :

For Annie, Annie's spending equation is Total spending = (2 * 10) + (2 * 2) + 6, Bob's spending equation is Total spending = (4 * 10) + (2 * 6), and Carol's spending equation is Total spending = (3 * 10) + 2.

For Bob, the total spending is calculated by multiplying the cost per kilo of apples (a) by 4 (since Bob buys 4 kilos of apples), and multiplying the cost of one coconut (c) by 2 (since Bob buys 2 coconuts). For Carol, the total spending is calculated by multiplying the cost per kilo of apples (a) by 3 (since Carol buys 3 kilos of apples), and adding the cost per kilo of bananas (b) (since Carol buys 1 kilo of bananas). The system of equations from part (a) in matrix form is:
[2 2 1] [a] [26]
[4 0 2] [b] = [40]
[3 1 0] [c] [18]

By multiplying the inverse of the coefficients matrix by the constants matrix,

we can solve for the values of a, b, and c. In this case, the values of a, b, and c are 10, 2, and 6 respectively.

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