Formula and Step by Step Solution of Cross Multiplication Method for 3 Variables
FAQs on Cross Multiplication Method for Solving 3 Variables and 3 Linear Equations
1. What is the cross multiplication method for 3 variables and 3 equations?
The cross multiplication method for 3 variables and 3 equations is a determinant-based technique used to solve a system of three linear equations using ratios of determinants. It is closely related to Cramer’s Rule.
- Consider the system:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃ - Find the determinant D of the coefficient matrix.
- Find Dₓ, Dᵧ, D𝓏 by replacing the respective columns with constants.
- The solution is:
x = Dₓ / D, y = Dᵧ / D, z = D𝓏 / D (if D ≠ 0).
2. What is the formula for solving 3 equations by cross multiplication?
The formula for solving three linear equations using cross multiplication is x = Dₓ/D, y = Dᵧ/D, z = D𝓏/D, where D is the determinant of the coefficient matrix.
- D = |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃|
- Dₓ = |d₁ b₁ c₁; d₂ b₂ c₂; d₃ b₃ c₃|
- Dᵧ = |a₁ d₁ c₁; a₂ d₂ c₂; a₃ d₃ c₃|
- D𝓏 = |a₁ b₁ d₁; a₂ b₂ d₂; a₃ b₃ d₃|
3. How do you solve 3 linear equations using cross multiplication method step by step?
To solve three linear equations using cross multiplication, compute determinants of the coefficient matrix and divide accordingly.
- Step 1: Write the system in standard form.
- Step 2: Form the coefficient determinant D.
- Step 3: Replace each variable’s column with constants to find Dₓ, Dᵧ, D𝓏.
- Step 4: Use x = Dₓ/D, y = Dᵧ/D, z = D𝓏/D.
4. Can you give an example of cross multiplication method for 3 variables?
Yes, for the system x + y + z = 6, x − y + z = 2, and x + y − z = 2, the solution by cross multiplication is x = 2, y = 2, z = 2.
- Coefficient determinant D = 4.
- Dₓ = 8, Dᵧ = 8, D𝓏 = 8.
- x = 8/4 = 2, y = 8/4 = 2, z = 8/4 = 2.
5. When can we use cross multiplication method for 3 equations?
The cross multiplication method can be used when the determinant of the coefficient matrix is non-zero, meaning D ≠ 0.
- If D ≠ 0 → unique solution exists.
- If D = 0 and Dₓ, Dᵧ, D𝓏 are also 0 → infinitely many solutions.
- If D = 0 but any of Dₓ, Dᵧ, D𝓏 ≠ 0 → no solution.
6. What is the determinant in cross multiplication method?
A determinant in the cross multiplication method is a numerical value calculated from the coefficients of variables arranged in a square matrix.
- For three variables, we use a 3×3 determinant.
- D = a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂).
7. What is the difference between Cramer’s Rule and cross multiplication method?
Cramer’s Rule and the cross multiplication method are essentially the same for solving linear equations using determinants.
- Both use determinant ratios: x = Dₓ/D, y = Dᵧ/D, z = D𝓏/D.
- Both require D ≠ 0 for a unique solution.
- “Cross multiplication method” is commonly used in school-level terminology.
8. Why does cross multiplication method fail when determinant is zero?
The cross multiplication method fails when the determinant is zero because division by zero is undefined.
- If D = 0, the expressions x = Dₓ/D, y = Dᵧ/D, z = D𝓏/D cannot be evaluated.
- This indicates either infinitely many solutions or no solution.
9. Is cross multiplication method suitable for all systems of 3 linear equations?
No, the cross multiplication method is suitable only for systems with a non-zero determinant.
- Works best for small systems like 3 variables and 3 equations.
- Becomes lengthy for larger systems.
- Requires D ≠ 0 for a unique answer.
10. What are common mistakes in cross multiplication method for 3 variables?
Common mistakes in the cross multiplication method include incorrect determinant expansion and wrong column replacement.
- Sign errors while expanding the 3×3 determinant.
- Replacing the wrong column when finding Dₓ, Dᵧ, or D𝓏.
- Forgetting to check if D = 0.
