Cross Multiplication Method
The cross-multiplication method is an algebraic technique used to solve a pair of linear equations in two variables. It involves cross-multiplying the coefficients and constant terms of the equations to directly find the values of the variables x and y. This method is highly efficient for finding a unique solution without the need to substitute or eliminate variables step-by-step.
Key Concepts
Before applying the method, both linear equations must be written in the standard form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
The method relies on a specific cyclic pattern of writing the coefficients of y, the constant terms, the coefficients of x, and again the coefficients of y.
The solution exists and the method is applicable only if the equations have a unique solution, meaning the ratio of a1 to a2 is not equal to the ratio of b1 to b2.
Real World Applications
Calculating the exact cost of individual items when given the total cost of two different combinations of items, such as finding the price of one pen and one notebook from two separate bulk purchases.
Determining the speed of a boat in still water and the speed of the river stream when given the time taken to travel specific distances upstream and downstream.