Cross-Multiplication Method for Linear Equations: Class 10

Quick answer: For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ with $a_1b_2 - a_2b_1 \ne 0$, the cross-multiplication method gives $\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$. For $2x + 3y = 46$ and $3x + 5y = 74$, this gives $\frac{x}{8} = \frac{y}{10} = \frac{1}{1}$, so $x = 8$ and $y = 10$.

Syllabus note: The cross-multiplication method (Section 3.4.3) was removed from the rationalised NCERT Class 10 textbook. Substitution and elimination remain in the syllabus. Cross-multiplication is still a quick check and is useful in competitive exams.

Standard form and types of solution

First write both equations with zero on the right: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.

Cross-multiplication gives the answer only in the first case, when $a_1b_2 - a_2b_1 \ne 0$.

The formula

$\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$

A memory aid

Write the coefficients in the order b, c, a, b in two rows (this order is sometimes remembered as "2 3 1 2", the positions of the coefficients in $ax + by + c$):

Each pair of neighbouring columns gives one denominator: multiply down-right, then subtract the product up-right.

• Columns b and c give $b_1c_2 - b_2c_1$, under $x$.
• Columns c and a give $c_1a_2 - c_2a_1$, under $y$.
• Columns a and b give $a_1b_2 - a_2b_1$, under 1.

Worked example

Solve $2x + 3y = 46$ and $3x + 5y = 74$ by cross-multiplication.

Step 1: Standard form: $2x + 3y - 46 = 0$ and $3x + 5y - 74 = 0$. So $a_1 = 2$, $b_1 = 3$, $c_1 = -46$, $a_2 = 3$, $b_2 = 5$, $c_2 = -74$.

Step 2: Check for a unique solution: $\frac{2}{3} \ne \frac{3}{5}$, so there is exactly one solution.

Step 3: Find the three denominators:

$b_1c_2 - b_2c_1 = 3(-74) - 5(-46) = -222 + 230 = 8$

$c_1a_2 - c_2a_1 = (-46)(3) - (-74)(2) = -138 + 148 = 10$

$a_1b_2 - a_2b_1 = 2(5) - 3(3) = 10 - 9 = 1$

Step 4: So $\frac{x}{8} = \frac{y}{10} = \frac{1}{1}$, giving $x = 8$ and $y = 10$.

Check: $2(8) + 3(10) = 46$ and $3(8) + 5(10) = 74$.

Answer: $x = 8$, $y = 10$.

The same example by elimination (in the current syllabus)

Step 1: Multiply the first equation by 3 and the second by 2: $6x + 9y = 138$ and $6x + 10y = 148$.

Step 2: Subtract: $y = 10$.

Step 3: Substitute: $2x + 30 = 46$, so $x = 8$.

Both methods give the same answer, which makes cross-multiplication a handy way to check your work.

Summary table

Exam tips

• Always bring the constants to the left so that $c_1$ and $c_2$ carry the correct signs.
• Write each product in brackets before subtracting to avoid sign slips.
• For CBSE board answers, use substitution or elimination unless your syllabus includes cross-multiplication.

Common mistakes

• Keeping the constants on the right-hand side and then using the formula unchanged, which flips the signs.
• Mixing up the order: the denominator under $y$ is $c_1a_2 - c_2a_1$, not $a_1c_2 - a_2c_1$.
• Writing $-222 - 230$ instead of $-222 - (-230)$.
• Using the method when $a_1b_2 - a_2b_1 = 0$, where there is no unique solution.

Frequently Asked Questions

What is the cross-multiplication formula?

x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁), for equations written in the form ax + by + c = 0.

When can the cross-multiplication method be used?

When a₁b₂ − a₂b₁ is not zero, which means the lines intersect and there is exactly one solution.

Is cross-multiplication in the CBSE Class 10 syllabus?

It was removed from the rationalised NCERT textbook. Substitution and elimination are the methods in the current syllabus.

What is the solution of 2x + 3y = 46 and 3x + 5y = 74?

x = 8 and y = 10.

What does the "2312" memory aid mean?

It is a way to remember the column order b, c, a, b, which are the 2nd, 3rd, 1st and 2nd coefficients of ax + by + c.