Cross Multiplication Method for 2 Variables

Dhristi JEE 2022-24

Cross Multiplication Method for 2 Variables Class 10

In Maths, we use the cross multiplication method for solving linear equations in two variables. This is the simplest method to solve them and gives us the accurate value of the two variables. However, the method of cross multiplication is applicable only when we have a pair of linear equations in two variables.


Cross Multiplication of Equations

Cross multiplication is used in arithmetic operations to solve the equations or for finding the value of the variable. In this process, we multiply the numerator of one side with the denominator of the other side. 


We have to multiply the numerator and denominator of one fraction by the denominator of the second fraction. Secondly, we can also multiply the numerator and denominator of the second fraction by the denominator of the first fraction. This will help to get an answer for the given question.


Consider that a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are two equations that need to be solved. By the method of cross-multiplication, we would find the values of the x and y variables.

x = \[\frac{b_{1}c_{2}-b_{2}c_{1}}{b_{2}a_{1}-b_{1}a_{2}}\] and y = \[\frac{c_{1}a_{2}-c_{2}a_{1}}{b_{2}a_{1}-b_{1}a_{2}}\]

Here, b2a1 - b1a2 ≠ 0

The final solution is,

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

This method given above is known as the ‘Cross-Multiplication Method’ as this technique is very useful when it comes to simplifying the solution. For memorizing the method of cross-multiplication and solving the linear equation in two variables, the diagram given below is helpful.

The arrows in the diagram indicate the multiplication of the values that are connected through the arrow. We can see that the second product is then subtracted from the first one. The result that we get is substituted as the denominator of the given variables and 1, as we can see above the arrow. The entire values that are obtained are then equated to form an equation as follows.

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

From here, we can evaluate the values of x and y, provided that b2a1 - b1a2 ≠ 0. Hence, we call this method cross-multiplication.In this method, the condition for the consistency of the given pair of linear equations in two variables has to be checked. These conditions are as follows:

  1. If \[\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\], the result is a unique solution and the given pair of the linear equations in two variables are known to be consistent.

  2. If \[\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}\] , there would be infinite solutions and the pair of the lines would be coincident and hence, consistent and dependent.

  3. If \[\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\], then there would be no existing solution and the pair of the given linear equations in two variables are inconsistent.


Solved Examples

Let us consider the following examples.

Example 1: Solve the following pair of linear equations by using the cross multiplication method.

3x − 4y = 2

y − 2x = 7

Solution: 

We can rewrite the above-given equation as:

3x − 4y = 2

−2x + y = 7

By the method of the cross multiplication,

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

When we substitute the values from the above given equation, 

\[\frac{x}{28+2}\] = \[\frac{y}{4+21}\] = \[\frac{1}{13-8}\]

\[\frac{x}{30}\] = \[\frac{x}{25}\] = -15

Hence, x = -6 and y = -5


Example 2: Determine the value of the given variables that satisfy the following equation: 2x + 5y = 20 and 3x+6y =12.

Solution: 

We have,

2x+5y = 20 

3x + 6y = 12

By the cross multiplication method, we know that

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

Hence, by substituting the values that we have in the above equation, we get,

\[\frac{x}{ [(5).(12) - (6).(20)] }\] = \[\frac{y}{ [(20).(3) - (12).(2)] }\] = \[\frac{1}{ [(5).(3) - (6).(2)] }\]

\[\frac{x}{(60-120)}\] = \[\frac{y}{(60-24)}\] = \[\frac{1}{(15-12)}\]

\[\frac{x}{(-60)}\] = \[\frac{y}{36}\] = \[\frac{1}{3}\]

On solving, we get,

\[\frac{x}{-60}\] = \[\frac{1}{3}\]

Hence, x = -20

And solving for y,

\[\frac{y}{36}\] = \[\frac{1}{3}\]

Hence, y = 12

Therefore, we have, x= -20 and y = 12, which is the point at which the given two equations would intersect.

FAQs on Cross Multiplication Method for 2 Variables

1. What is the cross multiplication method for solving linear equations?

An equation is a statement of two expressions that are equal. There can be one or more variables in an expression. When we find the variables, values that satisfy the given equation are called solving an equation. The cross multiplication method is used for finding the solution for a pair of equations having two variables. It is not used for linear equations in one variable.

2. How can we find the solution of linear equations in two variables using cross multiplication?

Students must understand the formula used for finding the solution of linear equations in two variables. The cross multiplication method is used for finding the solution of linear equations in two variables. The following equation can be used for finding the solution:

\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}\] = \[\frac{y}{c_{1}a_{2}-c_{2}a_{1}}\] = \[\frac{1}{b_{2}a_{1}-b_{1}a_{2}}\]

3. How can we get a unique solution in linear equations in two variables when using the cross multiplication method?

We can get a unique solution in linear equations in two variables when using the cross multiplication method if a1/a2≠b1/b2  and the pair of linear equations in two variables are consistent. This means that when we write the solution by using the cross multiplication method, and the term below 1 is non-zero, the solution will be unique.

4. What is the Rule of 3 Method in Mathematics?

The rule of 3 in terms of mathematics is an operation that helps us in quickly solving both the direct and the inverse proportion word problems. If we have to use the rule of 3, we must have three values: two values that are proportional to each other and a third value. From here, we can determine the value of the fourth one.


In simpler terms, the rule of three is a mathematical method in which we can determine the solution depending on the proportions. The best example of this is the cross-multiplication method wherein we find the values of the unknown variables by writing them in proportion.

5. What do you mean by the cross-multiplication method?

The cross-multiplication method is explained as follows.

In elementary algebra and elementary arithmetic, when there is an equation given between two fractions or two irrational numbers, we can cross-multiply the two for either simplifying the equation or finding out the values of the unknown variables. This method is referred to as the method of cross-multiplication. Practically, this method of cross-multiplication means that we need to multiply the numerator of one or each side of the given fraction by the denominator of the other side of the fraction, in a way that the terms tend to cross over.

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