Cross-multiplication method

Substitution and elimination both solve a pair of linear equations step by step, redoing the same arithmetic pattern every time. The cross-multiplication method skips the steps and writes the answer directly as a ratio built from the six coefficients of the pair — a formula rather than a procedure. It was once taught inside the chapter Pair of Linear Equations in Two Variables, but the current CBSE syllabus has dropped it: exams expect only the graphical, substitution and elimination methods. So why learn it at all? Because it is exactly what elimination produces when you carry it out once, symbolically, for any pair of coefficients — seeing that link makes elimination itself feel less like a trick and more like arithmetic. It also turns up in older textbooks and some competitive papers, so recognising the pattern is worth two minutes even though it will not appear on your CBSE answer sheet.

The core insight

For $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ , cross-multiplying the coefficients in a fixed diagonal pattern gives $x$ and $y$ directly as ratios, without repeating substitution or elimination from scratch.

The core idea

Write both equations in general form first — every term moved to one side, equal to zero:

a_1x + b_1y + c_1 = 0, \qquad a_2x + b_2y + c_2 = 0.

where: $a_1,b_1,c_1$ and $a_2,b_2,c_2$ are the coefficients of $x$ , $y$ and the constant term in each equation, and $x,y$ are the unknowns to be found. The method arranges these six numbers in a row, repeating $a_1,b_1$ at the end, and reads off diagonal products:

\begin{array}{ccccc} b_1 & c_1 & a_1 & b_1 & \\ b_2 & c_2 & a_2 & b_2 & \end{array}

Each of the three pairs of neighbouring columns gives one cross-product: multiply the downward-right diagonal, multiply the upward-right diagonal, subtract. That single sweep produces the $x$ -numerator, the $y$ -numerator, and the shared denominator in one pass:

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

where: the three expressions $b_1c_2-b_2c_1$ , $c_1a_2-c_2a_1$ and $a_1b_2-a_2b_1$ are each a difference of two diagonal products read straight off the array above. From this, $x = \dfrac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}$ and $y = \dfrac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}$ , valid whenever $a_1b_2-a_2b_1 \neq 0$ , which is exactly the condition for a unique solution. If that denominator is zero, the pair is dependent or inconsistent and cross-multiplication gives no single answer — the consistency ratio test ( $a_1/a_2$ vs $b_1/b_2$ vs $c_1/c_2$ ) has to be used instead. The appeal of the method is that it turns a two-step procedure (eliminate one variable, then back-substitute for the other) into a single lookup: once the array is written correctly, both $x$ and $y$ fall out of the same three products, with no back-substitution step at all. The cost is that the formula hides why it works, so a sign slip anywhere in the array is easy to make and hard to spot — which is precisely why substitution and elimination, where every intermediate step is visible and checkable, remain the syllabus methods for CBSE Class 10.

Figure

A repeating row of six coefficients b1 c1 a1 b1 over b2 c2 a2 b2 with three diagonal arrows showing which products form the x numerator the y numerator and the denominator

Derivation ADeriving the cross-multiplication formula from elimination

The goal is to show that the cross-multiplication ratios are not a separate rule to memorise but simply the elimination method carried out once, symbolically, on the general pair $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ , so that the finished formula can be trusted the same way a fully-checked elimination calculation would be trusted.

Step 1. Eliminate $y$ the usual way: multiply the first equation by $b_2$ and the second by $b_1$ , giving $a_1b_2x+b_1b_2y+b_2c_1=0$ and $a_2b_1x+b_1b_2y+b_1c_2=0$ . The $b_1b_2y$ terms are now identical in both.

Step 2. Subtract the second new equation from the first: $(a_1b_2-a_2b_1)x + (b_2c_1-b_1c_2) = 0$ , so $x = \dfrac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1}$ , exactly the ratio quoted above — elimination on $y$ has produced the $x$ -formula on its own.

Step 3. Eliminate $x$ by the mirror steps: multiply the first equation by $a_2$ and the second by $a_1$ , subtract, and simplify to get $y = \dfrac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}$ , matching the second ratio.

Step 4. Both results share the same denominator $a_1b_2-a_2b_1$ , which is why the method packages them as one three-part ratio $\dfrac{x}{b_1c_2-b_2c_1}=\dfrac{y}{c_1a_2-c_2a_1}=\dfrac{1}{a_1b_2-a_2b_1}$ instead of two separate formulas — and why it fails exactly when $a_1b_2-a_2b_1=0$ , the same condition that stalls elimination (no single value survives the subtraction).

Worked example

Solve using cross-multiplication: $2x+3y-7=0$ and $3x-2y-4=0$ .

Step 1. Identify coefficients: $a_1=2,\ b_1=3,\ c_1=-7$ and $a_2=3,\ b_2=-2,\ c_2=-4$ .

Step 2. Write the repeating array $3,\,-7,\,2,\,3$ over $-2,\,-4,\,3,\,-2$ and compute the denominator: $a_1b_2-a_2b_1 = (2)(-2)-(3)(3) = -4-9 = -13$ .

Step 3. Compute the $x$ -numerator: $b_1c_2-b_2c_1 = (3)(-4)-(-2)(-7) = -12-14 = -26$ .

Step 4. Compute the $y$ -numerator: $c_1a_2-c_2a_1 = (-7)(3)-(-4)(2) = -21-(-8) = -13$ .

Step 5. Form the ratios: $x = \dfrac{-26}{-13} = 2$ , $y = \dfrac{-13}{-13} = 1$ .

Step 6. Verify against both originals: $2(2)+3(1)-7 = 4+3-7 = 0$ and $3(2)-2(1)-4 = 6-2-4 = 0$ , both true. Answer: $x=2,\ y=1$ .

Competitive stretch

The same pair also solves cleanly by elimination, and comparing the two routes is instructive. Eliminating $y$ from $2x+3y=7$ and $3x-2y=4$ (multiply the first by $2$ , the second by $3$ , and add) gives $13x = 26$ , so $x=2$ — the identical value the array produced, in the identical number of multiplications, just organised differently. This is the concrete version of the derivation above: cross-multiplication is elimination pre-computed once for arbitrary coefficients, so any pair of linear equations you can eliminate, you can also cross-multiply, and the two answers must always agree.

Common traps

Mixing up the $x$ - and $y$ -numerator pattern. Always write out the full repeating array before reading off products rather than recalling the formula from memory alone.
Sign errors in the cross-products, especially when a coefficient is already negative — write each product with its sign in brackets before subtracting.
Forgetting the general-form requirement. All terms must be moved to one side ( $ax+by+c=0$ ) before reading off $a,b,c$ ; leaving an equation as $ax+by=c$ gives the wrong sign for $c$ .
Applying the formula when $a_1b_2-a_2b_1=0$ . This signals no unique solution — cross-multiplication cannot find a single point; use the consistency ratio test instead.
Reciting this as a CBSE exam method. It has been removed from the current syllabus, so an answer built on it may not match what a school evaluator is marking against, even though the numeric result is correct.

Ask the tutor: Want to see how the elimination method you use in exams turns into this cross-multiplication formula, or double-check a tricky sign in the coefficient array?

Why do I need to learn cross-multiplication if it is not on the CBSE syllabus?

You do not need it for a CBSE Class 10 exam answer, and using it there instead of substitution or elimination may not match what your evaluator expects even if the final numbers are correct. It is worth knowing anyway because it shows where the elimination method's answer comes from in general — the formula is just elimination done once and packaged for reuse, which deepens understanding of the method you do use, and it can save time in competitive exams where the technique is allowed.

How is cross-multiplication different from just doing elimination twice?

It is not really different — it is elimination done once, symbolically, for both $x$ and $y$ at the same time, with the answer stored as a ready-made formula. Elimination on a specific numerical pair repeats the multiply-and-subtract steps each time you solve a new pair; cross-multiplication skips straight to the ratios because those steps have already been carried out in general in the derivation above.

What if I get a zero in the denominator when using this method?

A zero denominator, $a_1b_2-a_2b_1=0$ , means the pair of equations does not have a single unique solution — the lines they represent are either parallel (no solution) or the same line (infinitely many solutions). Cross-multiplication cannot pick out one point in either case, so switch to comparing the ratios $a_1/a_2$ , $b_1/b_2$ and $c_1/c_2$ to decide which of the two situations you are in. This same zero-denominator condition also stalls elimination, since the derivation above shows both methods share it.

Cross-multiplication method

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads