Question

Two schools $A$ and $B$ want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award $₹ x$ each, $₹ y$ each and $₹ z$ each for the three respective values to $3, 2$ and $1$ students respectively with total award money of ₹ $1, 600$ . School $B$ wants to spend $₹ 2, 300$ to award its $4, 1$ and $3$ students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is $₹ 900$ , using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

Accepted Answer

Let the amount $x, y$ and $z$ be considered for sincerity, truthfulness and helpfulness.

According to the questions,

$3 x + 2 y + z = 1600$

$4 x + y + 3 z = 2300$

$x + y + z = 900$

Converting into the matrix form

$A X = B$

$[\begin{array}{l} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{matrix} 1600 \\ 2300 \\ 900 \end{matrix}]$

Applying row operations: $R_{1} \to R_{1} - 3 R_{3}$ and $R_{2} \to R_{2} - 4 R_{3}$

$[\begin{matrix} 0 & - 1 & - 2 \\ 0 & - 3 & - 1 \\ 1 & 1 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} - 1100 \\ - 1300 \\ 900 \end{matrix}]$

Applying row operations: $R_{2} \to 2 R_{2} - R_{1}$

$[\begin{matrix} 0 & - 1 & - 2 \\ 0 & - 5 & 0 \\ 1 & 1 & 1 \end{matrix}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{matrix} - 1100 \\ - 1500 \\ 900 \end{matrix}]$

Converting back into the equations we get

$- y - 2 z = - 1100$

$- 5 y = - 1500$

$x + y + z = 900$

$y = 300$

Putting value of $y$ in $- y - 2 z = - 1100$

$- 300 - 2 z = - 1100$

$- 2 z = - 800$

$z = 400$

Putting values in $x + y + z = 900$

$x + 300 + 400 = 900$

$x = 900 - 700$

$x = 200$

Hence, $₹ 200$ for sincerity, $₹ 300$ for truthfulness and $₹ 400$ for helpfulness. One more value may be like honesty, kindness, etc.

Important Questions on System of Linear Equations

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