Question

Use the Cramer's Rule to solve the word problem. Let x represent the number of cheeseburgers and y represent the number of bags of potato chips. Gopi loves cheeseburgers and bags of potato chips, and they have fat and calories as follows: cheeseburgers contain 18 grams of fat and 255 calories; bags of potato chips contain 19 grams of fat and 305 calories. How many cheeseburgers and bags of potato chips did Gopi eat when 166 grams of fat and 2495 calories were consumed?

Answer 1

Gopi consumed 5 bags of cheeseburgers and 4 bags of potatoes

Explanation:

x = number of bags of cheeseburgers

y = number of bags of potato chips

Cheeseburgers: 18 grams of fat, 255 calories

For x number of bags: 18x grams, 255x calories

Potato chips: 19 grams of fat, 305 calories

For y number of bags: 19y grams, 305y calories

Total fat consumed = 166

Total calories consumed = 2495

The equation for total fat:

18x + 19y = 166

The equation for total calories:

255x + 305y = 2495

Writing in matrix form:

$\begin{gathered} \begin{bmatrix}{18} & {19} & {} \\ {255} & {305} & {} \\ {} & {} & {}\end{bmatrix}\begin{bmatrix}{x} & {} & {} \\ {y} & & \\ {} & {} & \end{bmatrix}=\text{ }\begin{bmatrix}{166} & {} & {} \\ {2495} & {} & {} \\ {} & {} & {}\end{bmatrix} \\ \text{let A = }\begin{bmatrix}{18} & {19} & {} \\ {255} & {305} & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}$$\begin{gathered} A\det er\min ant\text{ =|A|} \\ \text{|A| = 18(305) - 19(255)} \\ \text{|A| = }5490-4845 \\ \text{|A| = 645} \end{gathered}$$\begin{gathered} A_x\text{ = }\begin{bmatrix}{166} & {19} & {} \\ {2495} & {305} & {} \\ {} & {} & {}\end{bmatrix}\text{ (using Cramer's rule)} \\ |A_x|\text{ = 166(305) - 19(2495)} \\ |A_x|\text{ =}50630\text{ - 47405} \\ |A_x|\text{ =}3225 \end{gathered}$$\begin{gathered} A_y\text{ = }\begin{bmatrix}{18} & {166} & {} \\ {255} & {2495} & {} \\ {} & {} & {}\end{bmatrix} \\ |A_y|\text{ = 18(2495) - 166(255)} \\ |A_y|\text{ = 44910 - 42330} \\ |A_y|\text{ =}2580 \end{gathered}$$\begin{gathered} x\text{ = }\frac{A_x}{|A|} \\ y\text{ = }\frac{A_y}{|A|} \\ \\ x\text{ = }\frac{3225}{645} \\ x\text{ = 5} \\ y\text{ =}\frac{\text{ 2580}}{645} \\ y\text{ = 4} \end{gathered}$

Hence, Gopi consumed 5 bags of cheeseburgers and 4 bags of potatoes