Cross Multiplication: Solving Linear Equation Two Variables

The first thing that one needs to know about the cross-multiplication formula Class 10 is that cross multiplication is a process that is used for simplifying the equations or for finding the value of a variable. This process is often seen in elementary arithmetic or algebraic simplification sums.

One can also define cross-multiplication as the process of multiplying the numerator of one side to the denominator of the other side. It can also be defined as the process of removing the fractions from an equation by multiplying on each side. This is done by a common multiple of the denominator of the fraction of both sides.

In the formula of cross multiplication, there is an important concept of the standard form. The standard form of the formula of cross multiplication method can represent the entire process in the form of:

A / b = c / d

This is an equation between two fractions. If we apply the cross multiplication method formula, then we will get ad = bc or a = bc / d. Usually, the cross multiplication method is primarily used for solving linear equations in two variables. Students should practice questions and solve the cross multiplication method.

When it comes to the question of what is cross multiplication and cross-multiplication rule, then students should also remember that this is the simplest method. This method yields an accurate value of the variables. It should also be noted that cross multiplication is only applicable when it comes to a pair of linear equations in two variables.

To further illustrate this point, let us assume that a₁x + b₁y + c₁ = 0 and a₂x + b₂x + c₂ = 0 are two equations. You need to solve these equations by using the cross multiplication method. To arrive at the answer and find out the values of x and y, we need to follow the steps that are mentioned below.

x = b₁c₂ - b₂c₁ / b₂a₁ - b₁a₂

y = c₁a₂ - c₂a₁ / b₂a₁ - b₁a₂

In these equations, b₂a₁ - b₁a₂ ≠ 0

Hence, the final solution is:

x / b₁c₂ - b₂c₁ = y / c₁a₂ - c₂a₁ = 1 / b₂a₁ - b₁a₂

The final solution of the simultaneous linear equation can be easily divided into two broad categories. These categories are:

• Graphical method

• Algebraic method

The algebraic method can be further classified into three divisions. These divisions are:

• Substitution method

• Elimination method

• Cross multiplication method

In this article, we will only be focused on the cross multiplication method. If you want to view a visual representation of this method, then you can also refer to the image that is attached below.

(Image will be uploaded soon)

The Purpose of Cross Multiplication Formula

By now, you must understand what cross multiplication means. This is why the next step in learning more about this topic is to understand the purpose of cross multiplication. Ideally, cross multiplication is used to simplify an equation. It can also be used to find the value of a variable in any given equation.

Students might also be interested to learn that cross multiplication is also used in subtraction and addition of unlike fractions. For example, if there is an equation of (20 / 2) = (a / 3) and we need to find the value of the variable ‘a’ by using the cross multiplication process, then we should follow the steps that are mentioned below.

20 x 3 = a x 2

60 = 2 a

a = 60 / 2

a = 30

Fun Facts About Cross Multiplication Method Class 10

Do you know that there is a specific derivation of the cross multiplication method? Let’s discuss this derivation now.

As a general rule, a pair of linear equations in two variables are represented as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

If we want to solve these pair of linear equations in two variables, then we have to follow some steps. And these steps are:

The given pair of linear equations in two variables are:

a₁x + b₁y + c₁ = 0                             ----(1)

a₂x + b₂y + c₂ = 0                             ----(2)

If the equation (1) is multiplied with b2 and equation (2) is multiplied with b1, then we will get:

b₂a₁x + b₂b₁y + b₂c₁ = 0              ----(3)

b₁a₂x + b₁b₂y + b₁c₂ = 0              ----(4)

Now, let’s subtract equation (4) from equation (3)

(b₂a₁ - b₁a₂)x + (b₂b₁ - b₁b₂)y + (b₂c₁ - b₁c₂) = 0

= (b₂a₁ - b₁a₂)x = b₁c₂ - b₂c₁

= x = b₁c₂ - b₂c₁ / b₂a₁ - b₁a₂

Here, it is given that b₂a₁ - b₁a₂ ≠ 0

After that, the value of x that was obtained has to either be substituted in equation (1) or equation (2). In this manner, we will be able to find the value of y:

y = c₁a₂ - c₂a₁ / b₂a₁ - b₁a₂

Hence, the solution of both the equations can be expressed as:

x / b₁c₂ - b₂c₁ = y / c₁a₂ - c₂a₁ = 1 / b₂a₁ - b₁a₂ -----(5)

The technique that is depicted in this derivation is known as the cross multiplication method. This technique can be used for simplifying various solutions and making it easier to memorize those solutions.

It should be noted that the arrows indicate the multiplication of the values that are connected through the arrows. After that, the second product is subtracted from the first product. The final result is later substituted as the denominator of the variables and 1. This is mentioned above the arrow and later the entire values are obtained by equating to form the equation (5).

x / b₁c₂ - b₂c₁ = y / c₁a₂ - c₂a₁ = 1 / b₂a₁ - b₁a₂

From this equation, x and y are evaluated. It is also provided that a1 b₂ - a₂b₁ ≠ 0. Students should remember that in this method, the condition for consistency of a pair of linear equations in two variables must be checked. This can be done by following the rules or tips mentioned below.

• If a₁ / a₂ ≠ b₁ / b₂, then that means that we will get a unique solution. Also, the pair of linear equations in two variables are completely consistent.

• If a₁ / a₂ = b₁ / b₂ = c₁ / c₂, then there are infinitely many solutions. And the pair of linear equations are coincident. This means that the equations are dependent and consistent.

• If a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂, then there are no solutions. Also, the pair of linear equations in two variables are also inconsistent.