Cross multiplication formula and steps with solved examples
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 to Solve Linear Equations in Two Variables
1. What is the cross multiplication method for solving linear equations in two variables?
The cross multiplication method is a technique used to solve a pair of linear equations in two variables by converting them into fractional form and equating cross products. It is mainly used for equations of the form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Using cross multiplication, we directly find:
- x / (b₁c₂ − b₂c₁) = y / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁)
It provides a quick algebraic method to calculate the values of x and y without elimination or substitution.
2. What is the formula for cross multiplication in linear equations?
The formula for the cross multiplication method is:
x / (b₁c₂ − b₂c₁) = y / (c₁a₂ − c₂a₁) = 1 / (a₁b₂ − a₂b₁)
For equations:
- a₁x + b₁y + c₁ = 0
- a₂x + b₂y + c₂ = 0
From this, we get:
- x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
- y = (c₁a₂ − c₂a₁) / (a₁b₂ − a₂b₁)
This formula works when the denominator (a₁b₂ − a₂b₁) ≠ 0.
3. How do you solve a pair of linear equations using cross multiplication step by step?
To solve a pair of linear equations using cross multiplication, first write both equations in standard form and then apply the formula directly.
Steps:
- Write equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
- Apply the formula:
x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁)
y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁) - Simplify the values.
Example:
2x + 3y − 11 = 0
4x − y − 5 = 0
Solving gives x = 2 and y = 1.
4. When can we use the cross multiplication method?
The cross multiplication method can be used when solving two simultaneous linear equations in two variables written in standard form.
- Both equations must be linear.
- They must be written as a₁x + b₁y + c₁ = 0.
- The value (a₁b₂ − a₂b₁) ≠ 0.
If (a₁b₂ − a₂b₁) = 0, the system may have no solution or infinitely many solutions.
5. What happens if a₁b₂ − a₂b₁ equals zero in cross multiplication?
If a₁b₂ − a₂b₁ = 0, the system of linear equations does not have a unique solution.
- If (b₁c₂ − b₂c₁) ≠ 0 → No solution (parallel lines).
- If (b₁c₂ − b₂c₁) = 0 and (c₁a₂ − c₂a₁) = 0 → Infinitely many solutions (coincident lines).
This condition helps determine whether the pair of equations is consistent or inconsistent.
6. Is cross multiplication the same as elimination method?
No, the cross multiplication method and the elimination method are different techniques for solving linear equations.
- Cross multiplication uses a direct formula involving determinants.
- Elimination removes one variable by adding or subtracting equations.
Both methods give the same result, but cross multiplication is often faster for standard-form equations.
7. Can you give an example of cross multiplication method?
Yes, here is a simple example of solving linear equations using cross multiplication.
Equations:
3x + 2y − 8 = 0
2x − 2y − 2 = 0
Using the formula:
- x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁)
- y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁)
After substitution and simplification, we get:
x = 2 and y = 1.
8. Why is cross multiplication method useful in solving linear equations?
The cross multiplication method is useful because it provides a direct formula to find x and y without multiple algebraic steps.
- Saves time in exams.
- Works efficiently for standard-form equations.
- Helps analyze consistency of equations.
It is especially helpful when coefficients are simple integers.
9. What are common mistakes in cross multiplication method?
Common mistakes in the cross multiplication method usually involve sign errors and incorrect arrangement of coefficients.
- Not writing equations in standard form.
- Mixing up the order of a₁, b₁, c₁ and a₂, b₂, c₂.
- Ignoring negative signs.
- Forgetting to check if (a₁b₂ − a₂b₁) ≠ 0.
Careful substitution and sign checking prevents most errors.
10. How is cross multiplication related to determinants?
The cross multiplication method is directly based on determinants of a 2×2 matrix.
- The denominator (a₁b₂ − a₂b₁) is the determinant of the coefficient matrix.
- The numerators are determinants formed by replacing columns with constants.
This concept is closely related to Cramer’s Rule, which also solves linear equations using determinants.
