Solve the following system of equations by the method of cross-multiplication:
$2x\ +\ y\ =\ 35$
$3x\ +\ 4y\ =\ 65$

Given:

The given system of equations is:

$2x\ +\ y\ =\ 35$

$3x\ +\ 4y\ =\ 65$

 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,

$2x+y-35=0$

$3x+4y-65=0$

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,

$a_1=2, b_1=1, c_1=-35$ and $a_2=3, b_2=4, c_2=-65$

Therefore,

$\frac{x}{1\times(-65)-(-35)\times4}=\frac{-y}{2\times(-65)-3\times(-35)}=\frac{1}{2\times4-3\times1}$

$\frac{x}{-65+140}=\frac{-y}{-130+105}=\frac{1}{8-3}$

$\frac{x}{75}=\frac{-y}{-25}=\frac{1}{5}$

$\frac{x}{75}=\frac{1}{5}$ and $\frac{-y}{-25}=\frac{1}{5}$

$x=\frac{75\times1}{5}$ and $-y=\frac{-25\times1}{5}$

$x=\frac{75}{5}$ and $-y=\frac{-25}{5}$

$x=15$ and $-y=-5$

$x=15$ and $y=5$

The solution of the given system of equations is $x=15$ and $y=5$.

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Updated on: 10-Oct-2022

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