Math Doubts

Cross multiplication method


$\dfrac{x}{b_{1}c_{2}-b_{2}c_{1}}$ $\,=\,$ $\dfrac{y}{c_{1}a_{2}-c_{2}a_{1}}$ $\,=\,$ $\dfrac{1}{a_{1}b_{2}-a_{2}b_{1}}$


$a_1{x}+b_1{y}+c_1 = 0$ and $a_2{x}+b_2{y}+c_2 = 0$ are a system of linear equations in two variables $x$ and $y$. In this case, $a_1$, $a_2$, $b_1$ and $b_2$ are coefficients of $x$ and $y$. $c_1$ and $c_2$ are constants in the pair of linear equations in two variables. The values of $x$ and $y$ can be calculated from the following formulas.

$(1) \,\,\,$ $x \,=\, \dfrac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}$

$(2) \,\,\,$ $y \,=\, \dfrac{c_{1}a_{2}-c_{2}a_{1}}{a_{1}b_{2}-a_{2}b_{1}}$

In both formulas, $a_1b_2-a_2b_1$ is a denominator. If it is equal to zero, then the values of $x$ and $y$ become infinity. So, it should not be equal to zero.

$a_1b_2-a_2b_1 \,\ne\, 0$

$\implies$ $a_1b_2 \,\ne\, a_2b_1$

$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{a_1}{a_2} \,\ne\, \dfrac{b_1}{b_2}$

Therefore, if $\dfrac{a_1}{a_2}$ is not equal to $\dfrac{b_1}{b_2}$, then the system of simultaneous linear equations has a unique solution.

cross multiplication method formula

In this method, the values of $x$ and $y$ can be written as the following equations by cross multiplication.

$\implies$ $\dfrac{x}{b_{1}c_{2}-b_{2}c_{1}} \,=\, \dfrac{1}{a_{1}b_{2}-a_{2}b_{1}}$

$\implies$ $\dfrac{y}{c_{1}a_{2}-c_{2}a_{1}} \,=\, \dfrac{1}{a_{1}b_{2}-a_{2}b_{1}}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{x}{b_{1}c_{2}-b_{2}c_{1}}$ $\,=\,$ $\dfrac{y}{c_{1}a_{2}-c_{2}a_{1}}$ $\,=\,$ $\dfrac{1}{a_{1}b_{2}-a_{2}b_{1}}$

This formula can be remembered easily as displaying in the diagram.

Due to the involvement of cross multiplication technique for writing this equation, the method of solving $x$ and $y$ is called the cross multiplication method.


Learn how to derive the formulas for the cross multiplication method.

Math Doubts

A best free mathematics education website that helps students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved