Systems of Equations

Linear systems

Two standard tools for a linear system in two variables:

• Substitution — solve one equation for one variable, plug into the other.
• Elimination — scale the equations so a variable's coefficients match (or are opposite), then add or subtract to cancel it.

A linear system can have exactly one solution, no solutions (parallel, inconsistent lines), or infinitely many (the same line written twice) — worth a half-second sanity check before you commit to solving.

Nonlinear systems

When one equation is a line and the other is a circle or parabola, substitution is almost always the move: solve the linear equation for one variable, substitute into the nonlinear one, and you're left with a single-variable equation (often quadratic).

The symmetry shortcut

When a system is symmetric in $x$ and $y$ — meaning it's built from$x+y$, $xy$, and $x^2+y^2$ rather than $x$ and $y$individually — it's often much faster to solve for the symmetric combinations directly using the identity $x^2+y^2 = (x+y)^2 - 2xy$, rather than solving for $x$ and $y$separately.

Practice problems