Last updated on July 5th, 2025
We use different methods to solve linear equations in math. One such method is cross multiplication, which is used to compare fractions or solve linear equations involving two variables. In this article, we will learn how to apply this method effectively.
The common method of multiplying numbers in the form of a fraction is known as cross multiplication. 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:
(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 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, either to determine the greater fraction or to check if they are equal. 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 10/1 (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:
(a)1x + (b)1y + (c)1 = 0
(a)2x + (b)2y + (c)2 = 0
The cross-multiplication formula we often apply is:
x(b1c1 - b2c2) = y(c1a2 - c2a1) = 1(b2a1 - b1a2)
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
(a)1x + (b)1y + (c)1 = 0…(1)
(a)2x + (b)2y + (c)2 = 0…(2)
We solve these equations by making the coefficients of y equal in both equations:
We first multiply equation (1) by b2 and equation (2) by b1:
(b2)(a)1x + (b2)(b)1y + (b2)(c)1 = 0…(3)
(b1)(a)2x + (b1)(b)2y + (b1)(c)2 = 0…(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(b1c1 - b2c2) = y(c1a2 - c2a1) = 1(b2a1 - b1a2)
Thus, we derived the required cross-multiplication formula.
Cross-multiplication is especially helpful in linear equations with two variables and is a quick way to find the solution. Solving linear equations with two variables, we apply the cross-multiplication method as given below:
Use the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 – b1a2)
Solve the linear equations:
3x + 4y = 8
2x + y = 5
Now, 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/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 – b1a2)
Substituting the values:
x/(4) (– 5) – (1) (– 8) = y/(– 8) (2) – (– 5) (3) = 1/ (1) (3) – (4) (2)
Simplifying the fractions:
x/(–20 + 8) = y/(– 16 + 15) = 1/ (3 – 8)
x/(–12) = y/(– 1) = 1/ (– 5)
Isolating x:
Comparing the first and third fractions:
x/ –12 = 1/–5
x = (–12 × 1)/ –5 = 12/5
Isolating y:
Compare the second and third fractions:
y/ – 1 =1/ –5
y = –1 × 1/ –5 = 1/5
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:
Solving linear equations using cross multiplication is a quick way to obtain results. However, students often make mistakes when solving equations using this method. Here are a few common mistakes and tips to avoid them:
Solve the following linear equations using the cross-multiplication method: 6x – 3y = 9 8x + 6y = 11
x = 29/20, y = –1/10
Rewrite the equations in the form ax + by + c = 0 (standard form):
6x - 3y - 9 = 0
8x + 6y -l 11 = 0
Now, we compare a1x+ b1y+ c1 = 0 and a2x + b2y + c2 = 0:
a1 =6, b1= –3, c1= -9
a2 = 8, b2 = 6, c2= -11
Apply the cross-multiplication formula:
x/(b1c2 – b2c1) = y/(c1a2– c2a1) = 1/(b2a1 – b1a2)
Substituting the values:
x/(–3 × –11– 6 × –9) = y/(– 9 × 8 – (– 11 × 6) = 1/ (6 × 6 – 8 × – 3 )
Simplify each term:
(–3 × –11) – (6 × –9) = 33 + 54 = 87
(–9 × 8) – (– 11 × 6) = –72 + 66 = – 6
(6 × 6) – (8 × –3) = 36 + 24 = 60
Thus, the equation simplifies to:
x/87 = y/ –6 = 1/60
Solve for x and y:
x = 87/60 = 29/20
y = –6/60 = –1/10
So, x = 29/20, y = –1/10
Solve the equation: 2/5 = x/ 10
x = 4
Apply the cross-multiplication method:
2 × 10 = 5 × x
20 = 5x
Solve for x by dividing both sides by 5:
x = 20/5
x = 4
Check if the fractions 3/12 and 5/20 are proportional.
3/12 and 5/20 are proportional (since both sides are equal.)
We use cross-multiplication to check if the given fractions are proportional:
Set up the proportion as:
3/12 = 5/20
Now apply cross-multiplication,
3 × 20 = 12 × 5
60 = 60
Since both sides are equal, we can confirm that 3/12 and 5/20 are proportional.
Anna needs 6 cups of flour to make 12 pancakes. How many cups of flour are needed to make 18 pancakes?
9 cups of flour.
Let the number of cups of flour required to make 18 pancakes be x
Set up the proportion:
6/12 = x/18
Now, apply cross multiplication:
6 × 18 = 12 × x
108 = 12x
Solve for x:
We divide both sides by 12:
x = 108/12 = 9
So, Anna needs 9 cups of flour to make 18 pancakes.
A store sells 9 apples for $7. How much would 20 apples cost?
The cost of 20 apples is approximately $15.56.
Let y be the cost of 20 apples,
We first set up the proportion:
9/7 = 20/ y
Apply cross multiplication:
9 × y = 7 × 20
9y = 140
y = 140/9 =15.556
So, the cost of 20 apples is approximately $15.56.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.