Simultaneous Equations (Copy)

5 Simultaneous Equations – Quick Reference Table

Type of System	Form	Key Method	Steps	Common Mistakes
Both Linear	ax + by = c dx + ey = f	Elimination OR Substitution	1. Align equations 2. Multiply to match coefficients 3. Add/subtract to eliminate one variable 4. Solve for the other	Forgetting to multiply both sides by same factor
One Linear, One Quadratic	e.g. y = 2x + 3 x² + y² = 25	Substitution	1. Sub linear into quadratic 2. Solve quadratic 3. Back-substitute for other variable	Dropping ± in square roots
Two Quadratics	e.g. x² + y² = r² y² – x² = k	Elimination OR Substitution	1. Add/subtract to remove squared term 2. Solve resulting equation 3. Find other variable	Not checking both equations for final values
Product Form	xy² = p xy = q	Division to simplify	1. Divide eqns to remove product 2. Solve for single variable 3. Use product eqn to find the other	Forgetting domain restrictions (no div by 0)
Fractional Form	(y/x) + 2y = m y = x – k	Substitution, then multiply through	1. Sub linear into fractional 2. Multiply through by denominator 3. Solve quadratic/linear	Not multiplying all terms by denominator
Mixed Variable Products	x² – 3xy + y² + c = 0 y – x + k = 0	Substitution for y in terms of x	1. Sub into second eqn 2. Expand + simplify 3. Solve quadratic	Sign errors in expansion

Quick Tips:

Always check both solutions in the original equations (some methods create extraneous roots).
For quadratic cases, expect up to two solution pairs.
When given fractions, state restrictions before multiplying (e.g. x ≠ 0).