Solving Nonlinear Systems: Step-by-Step Guide
A nonlinear system is a set of equations where at least one equation has a term that is not a straight line, like x^2, xy, or 1/x. The usual method is substitution: solve the simpler equation (often the line) for one variable, put it into the other equation, and solve. This often gives a quadratic equation. There can be 0, 1, 2, or more solutions.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Solve the easier equation for one variable. Pick the simpler equation, usually the linear one, and get one variable by itself.\[y = mx + b\]
- Put it into the other equation. Replace that variable in the second equation with what you found.\[f(x, mx+b) = 0\]
- Solve the new equation. You will usually get a quadratic. Solve it by factoring, completing the square, or using the quadratic formula.\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]
- Find the matching values. For each x you found, put it into the easy equation to find y. Each (x, y) pair is a possible solution.\[y_{i} = m x_{i} + b\]
- Check in both equations. Put each (x, y) pair into BOTH original equations. Throw out any that do not work in both.\[(x_{i}, y_{i}) \in \{\text{eq.1}\} \cap \{\text{eq.2}\}\]
Worked example
Solve the system: y = x + 1 and x^2 + y^2 = 25
Step 1: The first equation is already solved for y: y = x + 1.
Step 2: Put it into the second: x^2 + (x + 1)^2 = 25.
Step 3: Multiply out: x^2 + x^2 + 2x + 1 = 25, so 2x^2 + 2x - 24 = 0, which becomes x^2 + x - 12 = 0. Factor: (x + 4)(x - 3) = 0, so x = -4 or x = 3.
Step 4: Find y. If x = -4, y = -4 + 1 = -3. If x = 3, y = 3 + 1 = 4. The possible solutions are (-4, -3) and (3, 4).
Step 5: Check (-4, -3): -3 = -4 + 1 is true, and 16 + 9 = 25 is true. Check (3, 4): 4 = 3 + 1 is true, and 9 + 16 = 25 is true.
The solutions are (-4, -3) and (3, 4).
Tips and common mistakes
- Substitution is almost always the best method when one of the equations is not a line.
- (x + 1)^2 is x^2 + 2x + 1, not x^2 + 1. Do not skip the middle term.
- These systems can have more than one solution. Do not stop after finding the first.
- Always check in both original equations. Some answers can sneak in by mistake.
- If there is a square root or a fraction with x in it, make sure your answers are allowed.
- A picture can help. A line and a circle can cross in 0, 1, or 2 points.
- If the quadratic you get has no real solutions, the system has no real solutions.