The Power of Elimination

The Elimination Method is a technique used to solve a system of equations by adding or subtracting the equations to 'eliminate' one of the variables. This works because of the Addition Property of Equality: if you add equal amounts to both sides of an equation, the equation remains true. In a system, we look for variables with the same or opposite coefficients. For example, if one equation has $3x$ and the other has $-3x$, adding them together results in $0x$, effectively removing $x$ from the problem so we can solve for $y$.

What happens if the coefficients don't match? We use scaling. You can multiply an entire equation by a constant (any number) to force the coefficients to match. If you have $2x$ in one equation and $x$ in another, you can multiply the second equation by $2$ to get $2x$, or by $-2$ to get $-2x$. This is a powerful tool that allows us to solve any linear system, no matter how messy the numbers look initially.

In business, we often deal with two unknowns, such as the cost of different inventory items. We can translate these scenarios into systems. Let $x$ represent the quantity or price of item A, and $y$ represent item B. By setting up two equations—one for the total count and one for the total cost—we can use elimination to find the exact value of each item. This is the foundation of supply chain management and financial auditing.