How to Solve Systems of Equations
A system of linear equations involves a collection of two or more linear equations involving the same set of variables. In a standard two-variable system involving \( x \) and \( y \), solving the system means finding the exact mathematical coordinates where the two corresponding lines intersect on a graph. These intersecting points represent the unique values of \( x \) and \( y \) that make both equations mathematically true simultaneously.
The Substitution Method
The substitution method is highly effective when one of the variables in either equation is already isolated, or has a coefficient of 1. The procedural steps are:
- Algebraically manipulate one equation to solve for one variable (e.g., \( y = 5 - 2x \)).
- Substitute this newly formed expression into the second equation, entirely eliminating that variable.
- Solve the resulting single-variable equation, then substitute the numerical answer back into the first expression to find the remaining variable.
The Elimination Method (Addition Method)
When equations are written in the standard form \( Ax + By = C \), the elimination method is often preferred. The objective is to add or subtract the equations to temporarily eliminate one variable entirely.
To do this, you may need to multiply one or both equations by a constant so that the coefficients of either \( x \) or \( y \) become exact opposites (e.g., \( +3y \) and \( -3y \)). Once added vertically, the opposing terms cancel out to zero, leaving a straightforward equation with a single variable to solve.