Question

Combo 1Combo 2Combo 33 glazed5 glazed4 glazed4 cream filled6 cream filled4 cream filled5 chocolate1 chocolate4 chocolate$38$32$36a)Write a system to represent this situation. Use g for glazed donuts, f for cream filled donuts, and c for chocolate donuts.b)Solve the system ALGEBRAICALLY to find the price of each donuts.

Answer 1

We will use the following variables :

g for glazed

f for cream filled donuts

c for chocolate donuts

So, the equation for combo 1

3 g + 4 f + 5 c = $38

The equation for combo 2:

5 g + 6 f + c = $32

The equation for combo 3:

4 g + 4 f + 4 c = $36

So, the system of equations are:

3 g + 4 f + 5 c = 38 (1)

5 g + 6 f + c = 32 (2)

4 g + 4 f + 4 c = 36 (3)

B) Now, we need to solve the system of equations:

From equation 3:

4 g + 4 f + 4c = 36

divide all terms by 4

So, g + f + c = 9

Solve for c:

c = 9 - g - f

Substitute with the value of c at the equations (1)

At (1):

3 g + 4 f + 5 (9 - g - f) = 38

3g + 4f + 45 - 5g - 5f = 38

-2g - f = 38 - 45

-2g - f = -7

Multiply all terms by -1

2g + f = 7

Solve for f

f = 7 - 2g

Substitute with f at the equation of c

c = 9 - g - (7 - 2g)

c = 9 - g - 7 + 2g

c = g + 2

So, we have reached to :

f = 7 - 2g and c = g + 2

substitute with f and c at the equation (2)

5g + 6f + c = 32

5g + 6 (7 - 2g) + g + 2 = 32

solve for g

5g + 42 - 12 g + g + 2 = 32

5g - 12g + g = 32 - 42 - 2

-6g = -12

Divide both sides by -2

g = -12/-6 = 2

f = 7 - 2g = 7 - 2 * 2 = 7 - 4 = 3

c = g + 2 = 2 + 2 = 4

So, the cost of glazed = $2

The cost of cream filled = $3

The cost of chocolate = $4