Solve the following equations by reduction method :3x – y = 1, 4x… - Vidyadip

Question

Solve the following equations by reduction method $:3x – y = 1, 4x + y = 6$

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Answer

The given equations can be written in the matrix form as $:\left[\begin{array}{rr}3 & -1 \\ 4 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}1 \\ 6\end{array}\right]$
By $4 R_1$ and $3 R_2$, we get,
$\left[\begin{array}{rr}12 & -4 \\ 12 & 3\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{r}4 \\ 18\end{array}\right]$
By $R_2-R_1$, we get
$\left[\begin{array}{rr}12 & -4 \\ 0 & 7\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{r}4 \\ 14\end{array}\right] $
$\therefore \left[\begin{array}{r}12 x-4 y \\ 0+7 y\end{array}\right]=\left[\begin{array}{r}4 \\ 14\end{array}\right]$
By equality of matrices,
$12x – 4y = 4 … (1)$
$7y = 14 … (2)$
From $(2), y = 2$
Substituting $y = 2$ in $(1),$ we get,
$12x – 8 = 4$
$\therefore 12x = 12 $
$\therefore x = 1$
Hence, $x = 1, y = 2$ is the required solution.

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