Solve the following system of equations by the method of cross-multiplication:
$2x\ –\ y\ =\ 6$
$x\ –\ y\ =\ 2$

Given:

The given system of equations is:

$2x\ –\ y\ =\ 6$

$x\ –\ y\ =\ 2$

 To do: 

Here, we have to solve the given system of equations by the method of cross-multiplication.

Solution:  

The given system of equations can be written as,

$2x-y-6=0$

$x-y-2=0$

The solution of a linear pair(standard form) of equations $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ is given by,

$\frac{x}{b_1c_2-b_2c_1}=\frac{-y}{a_1c_2-a_2c_1}=\frac{1}{a_1b_2-a_2b_1}$

Comparing the given equations with the standard form of the equations, we get,

$a_1=2, b_1=-1, c_1=-6$ and $a_2=1, b_2=-1, c_2=-2$

Therefore,

$\frac{x}{-1\times(-2)-(-1)\times(-6)}=\frac{-y}{2\times(-2)-1\times(-6)}=\frac{1}{2\times(-1)-1\times(-1)}$

$\frac{x}{2-6}=\frac{-y}{-4+6}=\frac{1}{-2+1}$

$\frac{x}{-4}=\frac{-y}{2}=\frac{1}{-1}$

$\frac{x}{-4}=\frac{1}{-1}$ and $\frac{-y}{2}=\frac{1}{-1}$

$x=\frac{-4\times1}{-1}$ and $-y=\frac{2\times1}{-1}$

$x=\frac{-4}{-1}$ and $-y=\frac{2}{-1}$

$x=4$ and $-y=-2$

$x=4$ and $y=2$

The solution of the given system of equations is $x=4$ and $y=2$.

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Updated on: 10-Oct-2022

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