Step-by-Step Guide to Solving Two-Variable Equations with Cross Multiplication
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.
FAQs on Cross Multiplication Method for Solving Linear Equations in Two Variables
1. What exactly is the cross-multiplication method for solving linear equations?
The cross-multiplication method is an algebraic technique used to find the unique solution for a pair of linear equations in two variables. It provides a direct formula to calculate the values of the variables x and y by using the coefficients of the variables and the constant terms from both equations, which must be in the standard form (ax + by + c = 0).
2. What is the formula used in the cross-multiplication method?
For a pair of linear equations in the standard form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
The cross-multiplication formula to find the values of x and y is given by:
x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)
From this, you can find x by equating the first and last parts, and y by equating the second and last parts.
3. Can you provide a simple example of solving linear equations using the cross-multiplication method?
Certainly. Let's solve the system: 2x + y - 5 = 0 and 3x + 2y - 8 = 0.
Here, a₁=2, b₁=1, c₁=-5 and a₂=3, b₂=2, c₂=-8.
Using the formula:
x / [(1)(-8) - (2)(-5)] = y / [(-5)(3) - (-8)(2)] = 1 / [(2)(2) - (3)(1)]
x / [-8 + 10] = y / [-15 + 16] = 1 / [4 - 3]
x / 2 = y / 1 = 1 / 1
Now, we solve for x and y:
- For x: x/2 = 1 => x = 2
- For y: y/1 = 1 => y = 1
4. How is the cross-multiplication method different from the substitution or elimination methods?
The primary difference lies in the approach:
- Elimination/Substitution: These methods involve a series of algebraic steps to manipulate the equations, either by substituting one variable's value or by eliminating a variable by adding/subtracting the equations.
- Cross-Multiplication: This is a formula-based method. It directly provides the solution by plugging the coefficients into a predefined formula, which can be faster if you remember the formula correctly. However, it's more rigid and only works for a system of two equations.
5. Is the cross-multiplication method part of the CBSE Class 10 Maths syllabus for the 2025-26 board exams?
No, as per the latest NCERT curriculum and CBSE guidelines for the 2025-26 academic session, the cross-multiplication method has been removed from the 'Pair of Linear Equations in Two Variables' chapter for the Class 10 board examination. Students are expected to learn the substitution and elimination methods for solving these equations algebraically.
6. Can the cross-multiplication method be used to solve equations with three variables?
No, the standard cross-multiplication method is specifically designed to solve a system of two linear equations in two variables (like x and y). It cannot be directly applied to solve a system of linear equations with three variables (like x, y, and z). Solving for three or more variables requires more advanced techniques like Cramer's Rule or matrix methods (Gaussian elimination).
7. What are the conditions for unique, infinite, or no solutions in the cross-multiplication method?
The nature of the solution is determined by the coefficients, which can be checked using the formula's denominators:
- Unique Solution: The lines intersect at one point. The condition is a₁/a₂ ≠ b₁/b₂, which means the denominator (a₁b₂ - a₂b₁) is not zero.
- Infinitely Many Solutions: The lines are coincident. The condition is a₁/a₂ = b₁/b₂ = c₁/c₂.
- No Solution: The lines are parallel. The condition is a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
8. What is the main real-world application or importance of learning this method?
While removed from the syllabus, the cross-multiplication method is important for its connection to more advanced concepts in mathematics, particularly determinants and vectors. The formula's structure is a precursor to calculating a 2x2 determinant. In physics and engineering, the cross product of vectors, which is crucial for calculating torque and force, follows a very similar computational pattern. It teaches a systematic, formula-driven approach to problem-solving.
