1.1.4—Simultaneous equations

For two equations, eliminate one variable by substitution or a linear combination, solve the resulting equation, then back-substitute and verify in both original equations.

If one equation is quadratic, each valid root may produce a different pair. Keep every algebraic branch until the original equations or stated domain reject it.

From y=2x+1 and x²+y²=25, substitute to obtain x²+(2x+1)²=25; solve for both possible x values, then calculate y for each.

Solving only the positive square-root branch can omit a valid intersection, while numerical rounding can make a true pair appear not to satisfy the equations.