When graphical methods are imprecise (especially for fractional results), we use Algebraic Methods for exact solutions.
Substitution Method:
Best used when one variable is already isolated or has a coefficient of \(1\).
- Step 1: Express one variable in terms of the other from one equation (e.g., \(x = 3 - 2y\)).
- Step 2: Plug this expression into the other equation.
- Step 3: Solve for the remaining variable.
- Step 4: Substitute that value back into the Step 1 equation to find the first variable.
Elimination Method:
Best used when equations are in standard form (\(ax + by = c\)).
- Step 1: Multiply one or both equations by constants so that the coefficients of one variable are identical (or opposites).
- Step 2: Add or subtract the equations to "eliminate" that variable.
- Step 3: Solve the resulting one-variable equation.
- Step 4: Substitute the value into any original equation to find the other variable.
Important!
If the variables disappear during your calculations, look at the resulting statement:
- False Statement (e.g., \(0 = 5\)): The lines are parallel. No Solution.
- True Statement (e.g., \(0 = 0\)): The lines are coincident (the same line). Infinitely Many Solutions.