Cross Multiplication Method for 3 Variables and 3 Equations

To solve a system of equations like a₁x + b₁y + c₁z + d₁ = 0, a₂x + b₂y + c₂z + d₂ = 0, and a₃x + b₃y + c₃z + d₃ = 0, you arrange the coefficients in a specific repeating pattern and use the following formula derived from determinants:

x / [b₁(c₂d₃ - c₃d₂) - b₂(c₁d₃ - c₃d₁) + b₃(c₁d₂ - c₂d₁)] = -y / [a₁(c₂d₃ - c₃d₂) - a₂(c₁d₃ - c₃d₁) + a₃(c₁d₂ - c₂d₁)] = z / [a₁(b₂d₃ - b₃d₂) - a₂(b₁d₃ - b₃d₁) + a₃(b₁d₂ - b₂d₁)] = -1 / [a₁(b₂c₃ - b₃c₂) - a₂(b₁c₃ - b₃c₁) + a₃(b₁c₂ - b₂c₁)]

By equating the term for each variable with the constant term, you can solve for x, y, and z directly.

For the cross-multiplication method to yield a unique solution, the determinant of the coefficient matrix (the denominator for the constant term in the formula) must be non-zero. If this determinant is zero, it signifies that the system either has no solution (inconsistent) or infinitely many solutions (dependent), and this method cannot be used to find them.

The primary difference lies in complexity and the underlying structure.

• For 2 Variables: The method involves a simple cross-multiplication of four coefficients in a 2x2 arrangement. It's easy to memorise and apply.

• For 3 Variables: The method requires calculating 3x3 determinants, which is a more complex procedure. The visual "cross-multiplication" is not as straightforward and is replaced by a formula derived from the expansion of determinants (like Sarrus's rule or cofactor expansion).

While substitution and elimination are fundamental, the cross-multiplication method offers a different advantage. Its importance lies in providing a direct, formulaic solution without the need for sequential variable elimination. This can be faster and less prone to intermediate calculation errors if the formula is applied correctly, especially in scenarios where the coefficients are complex numbers or variables themselves.

They are very closely related but not identical in presentation. Cramer's Rule is the formal theorem that expresses the solution of a linear system in terms of ratios of determinants (Dx/D, Dy/D, Dz/D). The cross-multiplication method can be seen as a procedural shortcut or a mnemonic device that implements Cramer's Rule without the formal matrix notation, making it appear more like a formula.

No, not in its simple, formulaic form. The concept of using determinants (Cramer's Rule) can be extended to 4x4 systems and beyond. However, the specific pattern and calculation method (often taught as Sarrus's rule for 3x3 determinants) does not work for 4x4 or higher-order determinants. Calculating a 4x4 determinant requires a more complex method like Laplace expansion, making a simple "cross-multiplication" shortcut impractical and confusing.

Certainly. Consider the system:
x + y + z - 6 = 0
x - y + z - 2 = 0
2x + y - z - 1 = 0

Here, a₁=1, b₁=1, c₁=1, d₁=-6, and so on. You would substitute these coefficient values directly into the long-form cross-multiplication formula. For instance, the main determinant in the denominator would be calculated as:
1( (-1)(-1) - (1)(1) ) - 1( (1)(-1) - (1)(2) ) + 1( (1)(1) - (-1)(2) ) = 1(0) - 1(-3) + 1(3) = 6. Since the determinant is 6 (non-zero), a unique solution exists and can be found by calculating the numerators for x, y, and z.