Finding the intersection: how to solve simultaneous equations by elimination
In short: Imagine two moving paths that must meet at a single point. In algebra, simultaneous equations represent these paths. Solving them isn't just about finding numbers; it is about identifying the exact coordinate where two different rules coexist. The elimination method offers a streamlined way to strip away the noise.
Simultaneous equations involve two or more equations containing two or more unknowns, such as 'x' and 'y'. They are termed 'simultaneous' because the solution must satisfy all equations at the same time. Geometrically, if you plot these equations on a graph, the solution is the specific point where the lines cross. While there are three primary ways to find this intersection—graphing, substitution, and elimination—the elimination method is specifically designed for linear equations. These are equations where variables are not squared, meaning you won't find any x-squared or y-squared terms.
The mechanism of elimination relies on the principle of balancing. To begin, the equations must be organized so that the x-variables, y-variables, and constants are vertically aligned. The goal is to manipulate the equations so that one variable has the same coefficient in both. Once they match, you can eliminate that variable by adding or subtracting the two equations. A helpful mnemonic used in this process is 'Same Sign Subtract' (SSS): if the signs in front of the matching letters are the same, you subtract the equations to cancel them out. If the signs are different, you add them.
By successfully removing one variable, you are left with a single-variable equation that is much easier to solve. Once you find the value for the remaining letter, you substitute that number back into one of the original equations to reveal the value of the eliminated letter. The final result is an (x, y) coordinate pair. To ensure accuracy, it is essential to substitute both values back into the other original equation to verify that the solution holds true for both rules simultaneously.
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- Summary by Intellect · source: Simultaneous Equations By Elimination | Algebra | Maths | FuseSchool
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