When your equations are in the standard form $ax + by + c = 0$, the Cross-Multiplication Method is a powerful shortcut that allows you to calculate the values of $x$ and $y$ directly from their coefficients.
To use this method, always ensure your equations are set to zero:
$3x + 2y – 5 = 0$
$2x – 3y – 7 = 0$
Here, we identify our coefficients:
$a_1 = 3, b_1 = 2, c_1 = -5$
$a_2 = 2, b_2 = -3, c_2 = -7$
The cross-multiplication formula is organized as follows:
For $x$: $(2)(-7) – (-3)(-5) = -14 – 15 = -29$
For $y$: $(-5)(2) – (-7)(3) = -10 – (-21) = 11$
For the Constant (1): $(3)(-3) – (2)(2) = -9 – 4 = -13$
Our ratios now look like this:
Now we simply equate each variable ratio to the constant ratio:
Solving for $x$:
Solving for $y$:
Just like with the Elimination and Substitution methods, we find that:
This method is highly efficient because it eliminates the need for intermediate substitution steps, provided you are careful with your signs during the cross-multiplication!