Answer to Question #170942 in Math for Khan

Question

Write a system of linear equation representing any physical phenomena

(i) Solve the system by analytical method.

(ii) Solve the system by numerical method.

Answer 1

Suppose two high school kids during lunch break went to get some snacks. The first kid purchased two doughnuts and three sausage rolls and all cost $12. The other kid got two sausage rolls and a doughnut for $7. Assuming that all the snacks of equal size and amount, how much is the cost of a doughnut and a cost of sausage roll?

We can interpret the above statement as follows:

Let $x$ represents doughnut

Let $y$ represents sausage roll

Then for the first kid,

$$2x+3y =12 \qquad \cdots(i)$$

For the other kid,

$$2y+x= 7 \implies x+2y=7 \qquad \cdots (ii)$$

The above is, thus, a system of linear equation with two unknown variable.

(i) Solve the above system by analytical method.

From (ii),

$$x=7-2y \qquad \cdots(iii)$$

Substitute (iii) in (i)

$$2(7-2y)+3y =12\\ 14-4y+3y=12\\ 14-y =12\\ -y=12-14\\ -y =-2\\ y =2$$

Substitute $y=2$ in (iii)

$$x=7-2y\\ x=7-2(2)\\ x=7-4\\ x=3$$

Therefore, the cost of a doughnut is $3 while the cost of a sausage roll is $2.

(ii) Solve the above system by numerical method.

Using Gauss-Jordan Elimination Method,

The augmented matrix for the above system of equation is:

$$\left( \begin{array}{cc|c}2&3&12 \\\\ 1&2&7\end{array}\right)$$

We'll therefore make zeros in column 1 except the entry at row 1, column 1 (pivot entry).

Divide row 1 by 2, that is $\left(R_1=\frac{R_1}{2}\right)$ , we have:

$$\left[ \begin{array}{cc|c} 1 & \frac{3}{2} & 6 \\\\ 1 & 2 & 7 \end{array} \right]$$

Subtract row 1 from row 2, that is, $\left(R_2=R_2-R_1\right)$ , we have:

$$\left[ \begin{array}{cc|c} 1 & \frac{3}{2} & 6 \\\\ 0 & \frac{1}{2} & 1 \end{array} \right]$$

Make zeros in column 2 except the entry at row 2, column 2 (pivot entry).

Subtract row 2 multiplied by 3 from row 1, that is: $\left(R_1=R_1-\left(3\right)R_2\right)$ . We have:

$$\left[ \begin{array}{cc|c} 1 & 0 & 3 \\\\ 0 & \frac{1}{2} & 1 \end{array} \right]$$

Multiply row 2 by 2 , that is, $\left(R_2=\left(2\right)R_2\right)$ , we have:

$$\left[ \begin{array}{cc|c} 1 & 0 & 3 \\\\ 0 & 1 & 2 \end{array} \right]$$

From the above,

$$x=3, y=2$$

Therefore, the cost of a doughnut is $3 while the cost of a sausage roll is $2.