Name: Solution of Two Simultaneous Linear Equations
Uploaded: 2019-07-19
Duration: 9 min 55 s
Description: Two linear equations in 2 variables taken together are called simultaneous linear equations. The solution to these equations is the ordered pair which satisf...

Question

A total amount of

R s 7000

is deposited in three different saving bank accounts with annual interest rates

5 %

,

8 %

and

812 %

respectively. The total annual interest from these three accounts is

R s 550

.

Equal amounts have been deposited in the

5 %

and

8 %

saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

Accepted Answer

Given that A total amount of $R s 7000$ is deposited in three different saving bank accounts with annual interest rates $8 %$ , $8 %$ and $R s 550$ respectively. The total annual interest from these three accounts is $R s 550$ $.$ Equal amounts have been deposited in the $5 %$ and $8 %$ saving accounts.

Let the amount deposited in each of the three accounts be $R s x$ , and $R s y$ respectively.

Since, the total amount deposited is $R s 7, 000$ .

$∴$ $x + x + y = 7000$

$\Rightarrow 2 x + y = 7000 . . . . (1)$

Total annual Interest is $R s 550$ .

$∴$ $\frac{5}{100} x + \frac{8}{100} x + \frac{17}{200} y = 550$

$\Rightarrow 26 x + 17 y = 110000 . . . . (2)$

The above system of equations can be written in matrix form $A X = B$ as $[\begin{matrix} 2 & 1 \\ 26 & 17 \end{matrix}] (\binom{x}{y}) = (\binom{7000}{110000})$

Where $A = [\begin{matrix} 2 & 1 \\ 26 & 17 \end{matrix}], X = (\binom{x}{y})$ and $B = (\binom{7000}{110000})$

Now,

$|A| = |\begin{matrix} 2 & 1 \\ 26 & 17 \end{matrix}|$

$= 34 - 26$

$= 8$

Let $C_{i j}$ be the cofactors of elements $a_{i j}$ in $A = [a_{i j}]$ .

$C_{11} = {(- 1)}^{1 + 1} 17 = 17, C_{12} = {(- 1)}^{1 + 2} 26 = - 26$

$C_{21} = {(- 1)}^{2 + 1} 1 = - 1, C_{22} = {(- 1)}^{2 + 2} 2 = 2$

$a d j A = {[\begin{matrix} 17 & - 26 \\ - 1 & 2 \end{matrix}]}^{T}$

$= [\begin{matrix} 17 & - 1 \\ - 26 & 2 \end{matrix}]$

$\Rightarrow A^{- 1} = \frac{1}{|A|} a d j A$ $= \frac{1}{8} [\begin{matrix} 17 & - 1 \\ - 26 & 2 \end{matrix}]$

$X = A^{- 1} B$

$[\begin{matrix} x \\ y \end{matrix}] = \frac{1}{8} [\begin{matrix} 17 & - 1 \\ - 26 & 2 \end{matrix}] [\begin{matrix} 7000 \\ 110000 \end{matrix}]$

$\Rightarrow x = 1125 r s, y = 4750 r s$

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