Solve the following equations by reduction method :x + 3y = 2, 3x… - Vidyadip

Question

Solve the following equations by reduction method $:x + 3y = 2, 3x + 5y = 4.$

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Answer

The given equations can be written in the matrix form as $:\left[\begin{array}{ll}1 & 3 \\ 3 & 5\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}2 \\ 4\end{array}\right]$
By $R_2 – 3R_1$, we get
$\left[\begin{array}{lr}1 & 3 \\ 0 & -4\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left(\begin{array}{r}2 \\ -2\end{array}\right)$
$\therefore\left[\begin{array}{l}x+3 \\ 0-4 y\end{array}\right]=\left[\begin{array}{r}2 \\ -2\end{array}\right]$
By equality of matrices,
$\begin{array}{l}x+3 y=2 \ldots(1) \\ -4 y=-2\end{array}$
From $(2),y=\frac{1}{2}$
Substituting $y=\frac{1}{2}$ in $(1),$ we get,
$x+\frac{3}{2}=2$
$\therefore x=2-\frac{3}{2}=\frac{1}{2}$
Hence, $x=\frac{1}{2}, y=\frac{1}{2}$ is the required solution.

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