Solve the following equations by Cramer’s method. x+y-8 2 =x+2 y-… - Vidyadip

Question

Solve the following equations by Cramer’s method.
$\frac{x+y-8}{2}=\frac{x+2 y-14}{3}=\frac{3 x-y}{4}$

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Answer

Let,
$\frac{x+y-8}{2}=\frac{x+2 y-14}{3} $
$\Rightarrow 3 x+3 y-24=2 x+4 y-28$
$\Rightarrow x-y=-4 \ldots(1)$
Also,
Let $\frac{x+2 y-14}{3}=\frac{3 x-y}{4}$
$\Rightarrow 4 x+8 y-56=9 x-3 y$
$\Rightarrow 5 x-11 y=-56 \ldots(2)$
Hence the two equations are:
$x-y=-4 \ldots(1) $
$5 x-11 y=-56$
Now,
$D=\left|\begin{array}{cc}1 & -1 \\ 5 & -11\end{array}\right| $
$\Rightarrow D=(-11-(-5))=-6$
Also,
$ D _{ x }=\left|\begin{array}{rr} -4 & -1 \\-56 & -11\end{array}\right| $
$\text { Also, }$
$D _{ x }=44-56=-12$
And,
$D _{ y }=\left|\begin{array}{cc} 1 & -4 \\ 5 & -56 \end{array}\right| $
$\Rightarrow D _{ y }=-56+20=-36$
now, $ x=\frac{D_x}{D}=\frac{-12}{-6}=2 $
And, $y=\frac{D_y}{D}=\frac{-36}{-6}=6$
Hence, $(2,6)$ is the solution

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