Formula and Step by Step Procedure to Solve Two Linear Equations by Cross Multiplication Method
The concept of cross multiplication method for 2 variables plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you’re learning to solve algebraic equations for the first time or preparing for competitive examinations, mastering this reliable technique is essential for accuracy and speed.
What Is Cross Multiplication Method for 2 Variables?
The cross multiplication method for 2 variables is a systematic way to solve a pair of linear equations in two variables by multiplying and subtracting their coefficients in a specific pattern. You’ll find this concept applied when solving simultaneous equations, verifying solutions in algebraic expressions, and in shortcut strategies for competitive exams.
Key Formula for Cross Multiplication Method for 2 Variables
Here’s the standard formula for solving equations:
If you have:
\( a_1x + b_1y = c_1 \)
\( a_2x + b_2y = c_2 \)
| Variable | Formula | Notes |
|---|---|---|
| x | \( x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \) | Numerator and denominator use cross multiplication, and subtraction. |
| y | \( y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1} \) | Pattern follows similar crossing and subtraction. |
Cross-Disciplinary Usage
Cross multiplication method for 2 variables is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in questions involving systems of equations, circuit analysis, and even in solving real-life puzzles that require two unknowns.
Step-by-Step Illustration
Let’s solve the equations:
\( 3x - 4y = 2 \) (1)
\( y - 2x = 7 \) (2)
Equation (2) becomes: \( -2x + y = 7 \)
2. Identify the coefficients:
\( a_1 = 3,\ b_1 = -4,\ c_1 = 2 \)
\( a_2 = -2,\ b_2 = 1,\ c_2 = 7 \)
3. Cross multiply and subtract for x:
\( x = \frac{(-4) \times 7 - 1 \times 2}{3 \times 1 - (-2) \times -4} = \frac{-28 - 2}{3 - 8} = \frac{-30}{-5} = 6 \)
4. Cross multiply and subtract for y:
\( y = \frac{2 \times (-2) - 7 \times 3}{3 \times 1 - (-2) \times -4} = \frac{-4 - 21}{-5} = \frac{-25}{-5} = 5 \)
Final Answers: x = 6, y = 5
Speed Trick or Vedic Shortcut
Here’s a quick trick: Always align both equations in \( ax + by = c \) form, and remember the cross pattern with clear signs (−). Many students use a “diagram with arrows” to remind themselves of the order of multiplication and subtraction for both numerators and denominators.
Tip: The denominator in both x and y is always a₁b₂ − a₂b₁. If this denominator is zero, either the lines are parallel (no solution) or coincident (infinite solutions).
Try These Yourself
- Solve: \( 2x + 5y = 20 \), \( 3x + 6y = 12 \ )
- Solve: \( 5x - 3y = 7 \), \( 2x + 4y = 10 \ )
- If \( x + y = 8 \) and \( 2x - y = 3 \), find x and y by cross multiplication.
- Practice converting \( 2y - x = 4 \) into standard form before solving by cross multiplication method for 2 variables.
Frequent Errors and Misunderstandings
- Not arranging equations in standard \( ax + by = c \) form before applying the method.
- Forgetting to use the correct signs (minuses) in the numerator and denominator while cross multiplying.
- Swapping x and y numerators, leading to wrong answers.
- Applying cross multiplication when denominator is zero (system is inconsistent or dependent).
Relation to Other Concepts
The idea of cross multiplication method for 2 variables connects closely with the elimination method and the substitution method of solving simultaneous equations. Advanced learners will find this method similar to Cramer’s Rule for systems of equations and essential for progress in algebra, matrices, and even physics.
Classroom Tip
A quick way to remember the cross multiplication method for 2 variables is to sketch a big “X” connecting coefficients—top left to bottom right, top right to bottom left—and subtract as you go. Vedantu’s teachers often demonstrate this in live classes to help students memorize and apply the pattern without confusion.
Summary Table – Cross Multiplication Method for 2 Variables
| Step | Action |
|---|---|
| 1 | Write both equations in \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \) form |
| 2 | Calculate numerator for x: \( (b_1c_2 - b_2c_1) \ ) |
| 3 | Calculate numerator for y: \( (c_1a_2 - c_2a_1) \ ) |
| 4 | Denominator: \( a_1b_2 - a_2b_1 \ ) |
| 5 | Find \( x = \) numerator(x)/denominator, \( y = \) numerator(y)/denominator |
We explored cross multiplication method for 2 variables—from definition, formula, step-by-step solutions, mistakes, and its links to other solving methods. Continue practicing this reliable approach with Vedantu’s linear equations resources for maximum confidence in exams and real-life maths situations.
Related Reading: Linear Equations in Two Variables | Elimination Method | Simultaneous Equations | Cramer’s Rule
FAQs on Cross Multiplication Method for Pair of Linear Equations in Two Variables
1. What is the cross multiplication method for two variables?
The cross multiplication method is a shortcut technique used to solve a pair of simultaneous linear equations in two variables of the form ax + by + c = 0 and a₁x + b₁y + c₁ = 0. It directly gives the values of x and y using determinant-like ratios without eliminating variables step by step. This method is especially useful when both equations are written in standard form.
2. What is the formula for cross multiplication method?
The formula for the cross multiplication method is:
x / (b₁c − bc₁) = y / (ca₁ − ac₁) = 1 / (ab₁ − a₁b)
For equations:
ax + by + c = 0
a₁x + b₁y + c₁ = 0
From this, we get:
x = (b₁c − bc₁) / (ab₁ − a₁b)
y = (ca₁ − ac₁) / (ab₁ − a₁b)
This formula works when ab₁ − a₁b ≠ 0.
3. How do you solve simultaneous equations by cross multiplication method?
To solve simultaneous linear equations using the cross multiplication method, write both equations in standard form and apply the formula directly.
Steps:
- Write equations as ax + by + c = 0 and a₁x + b₁y + c₁ = 0.
- Apply: x / (b₁c − bc₁) = y / (ca₁ − ac₁) = 1 / (ab₁ − a₁b).
- Find x and y separately using the derived formulas.
Example:
2x + 3y − 5 = 0
4x − y − 1 = 0
x = (−1×−5 − 3×−1) / (2×−1 − 4×3) = (5 + 3) / (−2 − 12) = 8 / −14 = −4/7
y = ((−5×4) − (2×−1)) / (2×−1 − 4×3) = (−20 + 2) / −14 = −18 / −14 = 9/7
4. When can we use the cross multiplication method?
The cross multiplication method can be used when solving two linear equations in two variables written in standard form and when the determinant ab₁ − a₁b ≠ 0. If ab₁ − a₁b = 0, the system may have no solution or infinitely many solutions. This method is ideal for quick calculations in exams when coefficients are clearly identifiable.
5. Why does the cross multiplication method work?
The cross multiplication method works because it is derived from the concept of determinants and Cramer's Rule. The expression ab₁ − a₁b represents the determinant of the coefficient matrix. When this determinant is non-zero, the system has a unique solution. The cross products calculate determinants that directly give the values of x and y.
6. What is the difference between elimination method and cross multiplication method?
The elimination method removes one variable step by step, while the cross multiplication method directly applies a formula to find both variables.
- Elimination method: Add or subtract equations to eliminate one variable.
- Cross multiplication method: Use determinant-based formula to get x and y directly.
- Cross multiplication is faster for written standard-form equations.
Both methods give the same solution when applied correctly.
7. Can you give an example of cross multiplication method?
Yes, here is a simple example of solving equations using the cross multiplication method:
x + 2y − 4 = 0
3x − y − 5 = 0
Using the formula:
x = ((−1×−4) − (2×−5)) / (1×−1 − 3×2) = (4 + 10) / (−1 − 6) = 14 / −7 = −2
y = ((−4×3) − (1×−5)) / (1×−1 − 3×2) = (−12 + 5) / −7 = −7 / −7 = 1
So, the solution is x = −2 and y = 1.
8. What happens if ab₁ − a₁b = 0 in cross multiplication method?
If ab₁ − a₁b = 0, the system of equations does not have a unique solution. In this case:
- If (b₁c − bc₁) and (ca₁ − ac₁) are also zero, the equations have infinitely many solutions.
- If they are not zero, the equations have no solution.
This condition checks whether the lines are parallel or coincident.
9. Is cross multiplication method the same as Cramer's Rule?
Yes, the cross multiplication method is a simplified form of Cramer's Rule for two variables. Both methods use determinants to solve simultaneous linear equations. The expression ab₁ − a₁b is the determinant of coefficients, and the numerators are determinants formed by replacing coefficient columns with constants.
10. What are common mistakes in cross multiplication method?
Common mistakes in the cross multiplication method usually involve sign errors and incorrect pairing of coefficients.
- Not writing equations in standard form ax + by + c = 0.
- Making mistakes in cross products (wrong order).
- Forgetting negative signs while multiplying.
- Ignoring the condition ab₁ − a₁b ≠ 0.
Carefully following the formula and checking signs helps avoid calculation errors.
