How to Solve Simultaneous Equations (Elimination & Substitution Methods)

Simultaneous equations involve a set of two or more equations with the same group of unknown variables, where the goal is to find the specific values for these variables that satisfy *all* equations concurrently. They are fundamental in mathematics, crucial for solving real-world problems from economics to physics.

Two primary algebraic methods for solving them are substitution and elimination.

The **substitution method** operates by isolating one variable in one equation, expressing it in terms of the other. For instance, if you have `x + y = 5`, rearrange it to `y = 5 - x`. This expression is then substituted into the *other* original equation. This reduces it to a single equation with one unknown, which is solvable. After finding the first variable, substitute it back into an original equation to find the second.

The **elimination method** focuses on removing one variable. This is achieved by manipulating equations (often by multiplying them by constants) so that coefficients of one variable become either identical or additive inverses (e.g., 3y and -3y). Once coefficients are set, you add the equations (if inverses) or subtract them (if identical). This eliminates one variable, leaving a solvable single-variable equation. As with substitution, once one variable is found, substitute it back to determine the other.

Both methods reliably lead to the unique solution (if one exists), and the choice between them often depends on which appears simpler given the initial form of the equations.