Cross Multiplication Method for Linear Equations

Cross Multiplication Method for Linear Equations converts an equation involving fractions into a simpler linear equation. Multiply each numerator by the denominator on the opposite side, expand the brackets, collect the variable terms and solve. The method applies only when the denominators are non-zero.

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What Is the Cross Multiplication Method for Linear Equations?

The cross multiplication method solves a proportion by multiplying opposite numerator-denominator pairs. For a/b = c/d, cross multiplication gives ad = bc, provided b and d are non-zero.

For a fractional linear equation, first identify the two sides of the equality and multiply diagonally. For example, (2x + 3)/5 = 7/10 gives 10(2x + 3) = 5 × 7. Expanding gives 20x + 30 = 35, so x = 1/4. A denominator containing x must also be checked after solving.

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Simultaneous Equations Cross-Multiplication

Learn how to solve simultaneous linear equations using the cross-multiplication method, including the required formula, substitution of coefficients, and calculation of both unknowns.

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Cross Multiplication Method for Linear Equations Formula & Tricks

Important Formulas

Basic cross multiplication rule
a/b = c/d ⟹ ad = bc, where b ≠ 0 and d ≠ 0

Multiply the numerator on one side by the denominator on the opposite side.

Linear fractional equation
(ax + b)/(cx + d) = p/q ⟹ q(ax + b) = p(cx + d)

The denominators must be non-zero: cx + d ≠ 0 and q ≠ 0.

Direct solution formula
x = (pd − qb)/(qa − pc), provided qa − pc ≠ 0

This follows by expanding q(ax + b) = p(cx + d) and collecting the x-terms.

Quick Tricks

Write the diagonal products clearly

For A/B = C/D, use A × D = B × C. The first numerator is paired with the second denominator, and the second numerator is paired with the first denominator.

Example: (3x − 2)/4 = 5/6 gives 6(3x − 2) = 4 × 5, not 4(3x − 2) = 6 × 5.
Handle negative signs before expanding

Keep the negative sign attached to its complete numerator or denominator. A negative denominator may be moved to the numerator by changing the sign.

Example: (x + 1)/(−3) = 4/5 gives 5(x + 1) = −12, so x = −17/5.
Check denominator restrictions

Cross multiplication is valid only when every original denominator is non-zero. Reject any calculated value that makes a denominator zero.

Example: In (x + 2)/(x − 3) = 2, x = 3 is not allowed because the original denominator becomes zero.

Cross Multiplication Method for Linear Equations Concepts

Solving a proportion with constants

For a/b = c/d, multiply diagonally to obtain ad = bc and then simplify.

This form contains no variable in the denominators. After cross multiplication, move constants to isolate the unknown. Both original denominators must be non-zero.

Example: For (x + 2)/3 = 5/6, 6(x + 2) = 15. Thus 6x + 12 = 15, so x = 1/2.

Solving a fractional linear equation

For (ax + b)/(cx + d) = p/q, cross multiplication produces q(ax + b) = p(cx + d).

Expand both sides, bring all x-terms to one side and constants to the other side. The restriction cx + d ≠ 0 must be applied before accepting the answer.

Example: For (2x + 1)/(x + 3) = 3/4, 4(2x + 1) = 3(x + 3). Hence 8x + 4 = 3x + 9, so x = 1. The denominator is 1 + 3 = 4, which is non-zero.

Sign handling in cross multiplication

The signs of the two diagonal products must be retained exactly while cross multiplying.

If one fraction is negative, one diagonal product is negative. If both fractions are negative, the resulting equality has positive diagonal products after the signs are simplified.

Example: For (x − 4)/7 = −2/3, 3(x − 4) = −14. Therefore 3x − 12 = −14 and x = −2/3.

Special cases after cross multiplication

If the coefficient of x becomes zero, the equation may have no solution or infinitely many solutions.

After cross multiplication, an equation such as 0x = a with a ≠ 0 has no solution. An equation such as 0x = 0 is true for every value allowed by the original denominators.

Example: The equation (x + 1)/(x + 2) = (x + 1)/(x + 2) is true for every x except x = −2, because the denominator cannot be zero.

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Practice Cross Multiplication Method for Linear Equations Questions

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1Cost price of an article is Rs. 4800. The loss percentage is 40 percent. What is the selling price of the article?→ 2A trader sold goods for ₹1250 at a profit of 25% . What was the cost price?→ 3Vipul's salary was first reduced by 40% and subsequently raised by 46%. How much percentage was his final salary lower compared to his initial salary?→ 4Simplify [8.16/2.4+(3.64 × 1.5)] to two decimal places.→ 5If 20% of x = y, then y% of 20 is the same as:→ 6The cost price of 60 oranges is equal to the selling price of 25 oranges. Find the profit percentage.→ 7A shopkeeper bought certain goods for ₹2,500 and sold them for ₹3,250. Find his profit percentage.→ 8Two successive discounts of 25 percent and 20 percent are given on an article. What is the net discount?→ 9√1156 + √0.36 + √0.0025 = ?→ 10A trader marks his goods 50% above the cost price and allows a discount of 10% on it. If his gain is x%, find the value of x.→

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Quick Revision Notes

Cross Multiplication Method: Revision Points

Use these rules to solve and verify fractional linear equations.

  • For a/b = c/d, use ad = bc.
  • For (ax + b)/(cx + d) = p/q, use q(ax + b) = p(cx + d).
  • Expand brackets only after forming the two diagonal products.
  • Move variable terms to one side and constants to the other side.
  • Check that every original denominator is non-zero.
  • If qa − pc ≠ 0, then x = (pd − qb)/(qa − pc).
  • If cross multiplication gives 0x = a, a ≠ 0, there is no solution.
  • If cross multiplication gives 0x = 0, all values satisfying denominator restrictions are solutions.

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Cross Multiplication Method for Linear Equations FAQs

What is the cross multiplication formula for linear equations?

For (ax + b)/(cx + d) = p/q, the formula is q(ax + b) = p(cx + d). If qa − pc ≠ 0, then x = (pd − qb)/(qa − pc).

How do you solve (3x + 2)/5 = 4/7?

Cross multiply: 7(3x + 2) = 5 × 4. Thus 21x + 14 = 20, so x = −14/21 = −2/3.

Can cross multiplication be used when a denominator is zero?

No. Cross multiplication requires all original denominators to be non-zero. Any value that makes a denominator zero must be rejected.

How is a negative denominator handled in cross multiplication?

Retain the negative sign in the diagonal product or transfer it to the numerator. For example, (x + 1)/(−4) = 3/5 gives 5(x + 1) = −12.

What is the solution of (4x − 1)/(x + 2) = 3/2?

Cross multiplication gives 2(4x − 1) = 3(x + 2). Therefore 8x − 2 = 3x + 6, so 5x = 8 and x = 8/5. The denominator is 18/5, so the solution is valid.

What happens if the coefficient of x is zero after cross multiplication?

If the result is 0x = a with a non-zero, there is no solution. If the result is 0x = 0, every value satisfying the original denominator restrictions is a solution.

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