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Answer:$\mathrm{x}=5-2 \mathrm{k}, \mathrm{y}=\mathrm{k}$

Hint: Use Cramer’s rule for system of linear equations.

Given:

\begin{aligned} &x+2 y=5 \\ &3 x+6 y=15 \end{aligned}

Solution:

Solving determinant,

$|\mathrm{A}|=\left|\begin{array}{ll} 1 & 2 \\ 3 & 6 \end{array}\right| \quad \because\left|\begin{array}{ll} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array}\right|=\left(a_{1} b_{2}-a_{2} b_{1}\right)$

\begin{aligned} &=6-6 \\ &=0 \end{aligned}

|A| = 0 $\Rightarrow$System of linear equations have infinite number of solutions.

Now for x, the x column is replaced with constant column i.e.

$\Rightarrow \mathrm{D}_{x}=\left|\begin{array}{cc} 5 & 2 \\ 15 & 6 \end{array}\right|=30-30=0$

If we are solving for y, the y column is replaced with constant column i.e.

$\Rightarrow \mathrm{D}_{\mathrm{x}}=\left|\begin{array}{cc} 1 & 5 \\ 3 & 15 \end{array}\right|=15-15=0$

$\text { Since, } \mathrm{D}=\mathrm{D}_{\mathrm{x}}=\mathrm{D}_{\mathrm{x}}=0$

Let y = k, then we have:

\begin{aligned} &\Rightarrow \mathrm{x}+2 \mathrm{k}=5 \\ &\Rightarrow \mathrm{x}=5-2 \mathrm{k} \end{aligned} are the infinitive solutions of the given system.

Concept: Solving matrix of order 2x2 by Cramer’s rule.

Note: When D = 0, there is either no solution or infinite solutions.

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