Cross Multiply

Cross multiplication is a common method of multiplying numbers in fraction form. Cross multiplication involves multiplying the numerators of one fraction by the denominator of the other across an equation. This method can be easily applied when solving linear equations with two variables. For example:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

If cross-multiplication is performed correctly, we can quickly obtain the values of x and y by simplifying the equation.
 

Cross multiplication is primarily used with fractions to compare them or check their equality. This method can help solve equations when dealing with complex fractions. Given below are the different steps involved in this process:

For example: 5/8 = 10/16

Step 1: To begin, multiply the fraction's numerator on the right by the fraction's denominator on the left. Multiplying 10 × 8 = 80.

Step 2:  Now, we multiply the fraction's denominator on the right by the fraction's numerator on the left. We can represent 10/16 as 5/8 (since they are equivalent fractions).

 
Step 3: After the cross multiplication, always compare the LHS and RHS.
If they are equal, we can conclude that the fractions are equivalent. Here, in this 80 = 80, the fractions are equivalent.

The cross-multiplication formula is used for solving linear equations with two variables, as given below: 
 

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
 

The cross-multiplication formula used to solve linear equations is:

\({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)

How to Derive Cross-Multiplication Formula?

In linear equations with two variables, we derive the cross-multiplication formula by eliminating one variable, often by making the coefficients of that variable equal.

For example: Let’s consider two linear equations as

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

We solve these equations by making the coefficients of y equal in both equations:
 

• We first multiply all terms of equation (1) by b₂ and all terms of equation (2) by b₁:
  
  b₂ × (a₁x + b₁y + c₁) = 0
  ⇒ b₂a₁x + b₂b₁y + c₁....(3)
  
  b₁ × (a₂x + b₂y + c₂) = 0
  ⇒ b₁a₂x + b₁b₂y + b₁c₂…(4)
  
   

• Then, subtract equation (4) from equation (3), 
  
  (b₂a₁ – b₁a₂)x + (b₂c₁ – b₁c₂) = 0
  
   
• Isolating x,
  
  x = (b₁c₂ – b₂c₁) / (b₂a₁ – b₁a₂)…(a)
  
  where (b₂a₁ – b₁a₂) ≠ 0
  
   

• Similarly, solve (1) and (2) for y:
  
  y = (c₁a₂ – c₂a₁) / (b₂a₁ – b₁a₂)…(b)
  
  where (b₂a₁ – b₁a₂) ≠ 0
  
   
• We now combine (a) and (b),
  
  \({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)

Thus, we derived the required cross-multiplication formula.

Cross-multiplication is a quick and effective method for solving linear equations with two variables. Solving linear equations with two variables, we apply the cross-multiplication method as given below:

Use the cross-multiplication formula:

 
\({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)
 

Solve the linear equations:

3x + 4y = 8
2x + y = 5

 

• First, we convert the equations into standard form:
  Since the general form of a linear equation is:
  
  ax + by + c = 0
  
  We rewrite the given equation as:
  
  3x + 4y – 8 = 0
  2x + y – 5 = 0
  
   
• When comparing the general form: a₁x + b₁y = – c₁ and a₂x + b₂y = – c₂:
  
  a₁ = 3, b₁ ​= 4, c₁ ​= −8
  a₂ = 2, b₂ = 1, c₂ = −5
  
   
• Now, apply the cross-multiplication formula:
  
  \({x \over b_1c_2 \space –\space b_2c_1} = {y \over c_1a_2 \space– \space c_2a_1} = {1 \over b_2a_1 \space–\space b_1a_2}\)
  
   
• Substituting the values:
  
  \({x \over (4\times-5) \space –\space (1\times-8)} = {y \over (-8\times2) \space– \space (-5\times3)} = {1 \over (1\times 3) \space–\space (4\times2)}\)
  
   
• Simplifying the fractions:
  
  \({x \over (-20) \space –\space (-8)} = {y \over (-16) \space– \space (-15)} = {1 \over 3 \space–\space 8}\)

  
  \({x \over (-12)} = {y \over (-1)} = {1 \over (-5)}\)

  
   

   

• Isolating x:
  Comparing the first and third fractions:
  
  \({x \over (-12)} = {1 \over (-5)}\)
  
  \({x } = {{-12 \times 1} \over (-5) } = {12 \over 5}\)
  
   
• Isolating y:
  Compare the second and third fractions:
  
  \({y \over (-1)} = {1 \over (-5)}\)
  
  \({y } = {1 \times -1 \over (-5)} = {1 \over 5}\)
  
  
  ⇒ x = 12/ 5, y = 1/5

Cross-multiplication in equations helps students determine the value of unknown numbers easily. This technique can be applied to various real-life situations. Here are a few examples of its applications:

• Cross-multiplication is used to compare the prices of different items for choosing the cheaper option.
   
• This method helps adjust the quantities of ingredients required for a recipe without changing the proportions.
   
• Cross-multiplication is utilized in calculating the time required for a journey at a constant speed.
   
• We apply cross-multiplication to convert currencies accurately on the basis of their exchange rates.