🎯 Systems of Equations — Linear & Nonlinear
Essential for word problems and geometric applications in AMC contests.
🎯 Key Ideas
Linear Systems — Multiple linear equations with multiple variables, solved by substitution or elimination. The number of solutions depends on the relationship between equations.
Nonlinear Systems — Systems involving quadratics, circles, or other curves, often solved by substitution. These frequently appear in geometric problems.
Parameter Analysis — Understanding how the number of solutions depends on parameter values, often using discriminant or geometric intuition.
📊 Solution Methods
Linear Systems (2×2)
| Method | When to Use | Example |
|---|---|---|
| Substitution | One variable easily isolated | $y = 2x + 1$ into $3x + 2y = 7$ |
| Elimination | Coefficients are convenient | Add/subtract equations to eliminate variable |
| Graphing | Visual understanding needed | Find intersection points |
Linear Systems (3×3)
| Method | When to Use | Example |
|---|---|---|
| Gaussian Elimination | Systematic approach | Row operations to triangular form |
| Substitution | One equation is simple | $z = 2$ from $z = 2$ |
| Cramer’s Rule | Determinants are easy | Use formula for each variable |
Nonlinear Systems
| Method | When to Use | Example |
|---|---|---|
| Substitution | One equation is linear | $y = x + 1$ into $x^2 + y^2 = 5$ |
| Elimination | Both equations are similar | Subtract to eliminate terms |
| Symmetry | Equations have symmetry | $x^2 + y^2 = 1$ and $xy = \frac{1}{2}$ |
🎯 Micro-Examples
Example 1: Solve $\begin{cases} 2x + y = 7 \ x - y = 2 \end{cases}$
- Elimination: Add equations: $3x = 9$, so $x = 3$
- Substitute: $3 - y = 2$, so $y = 1$
- Answer: $(3, 1)$
Example 2: Solve $\begin{cases} x^2 + y^2 = 25 \ x + y = 7 \end{cases}$
- Substitution: $y = 7 - x$ from second equation
- Substitute: $x^2 + (7-x)^2 = 25$
- Expand: $x^2 + 49 - 14x + x^2 = 25$
- Simplify: $2x^2 - 14x + 24 = 0$, so $x^2 - 7x + 12 = 0$
- Factor: $(x-3)(x-4) = 0$, so $x = 3, 4$
- Answer: $(3, 4)$ and $(4, 3)$
Example 3: Solve $\begin{cases} x + y + z = 6 \ 2x - y + z = 3 \ x + 2y - z = 4 \end{cases}$
- Elimination: Add first two: $3x + 2z = 9$ (equation A)
- Elimination: Add first and third: $2x + 3y = 10$ (equation B)
- Solve: From A: $z = \frac{9-3x}{2}$; from B: $y = \frac{10-2x}{3}$
- Substitute: $x + \frac{10-2x}{3} + \frac{9-3x}{2} = 6$
- Solve: $x = 1$, so $y = \frac{8}{3}$, $z = 3$
- Answer: $(1, \frac{8}{3}, 3)$
⚠️ Common Traps & Fixes
Trap: Forgetting to check all solutions
- Fix: Always substitute back into original equations
- Example: $(3, 4)$ and $(4, 3)$ both satisfy $x^2 + y^2 = 25$ and $x + y = 7$
Trap: Dividing by zero in elimination
- Fix: Check if coefficients are zero before dividing
- Example: If $x$ coefficient is zero, use different elimination strategy
Trap: Extraneous solutions in nonlinear systems
- Fix: Squaring both sides can introduce extra solutions
- Example: $x + y = 7$ and $x^2 + y^2 = 25$ gives $(3,4)$ and $(4,3)$, but $(4,3)$ might not satisfy original if there were restrictions
🎯 AMC-Style Worked Example
Problem: Find all real solutions to the system: $$\begin{cases} x^2 + y^2 = 13 \ xy = 6 \end{cases}$$
Solution:
- Strategy: Use substitution since both equations are symmetric
- From second equation: $y = \frac{6}{x}$ (assuming $x \neq 0$)
- Substitute: $x^2 + \left(\frac{6}{x}\right)^2 = 13$
- Simplify: $x^2 + \frac{36}{x^2} = 13$
- Multiply by $x^2$: $x^4 + 36 = 13x^2$
- Rearrange: $x^4 - 13x^2 + 36 = 0$
- Substitute $u = x^2$: $u^2 - 13u + 36 = 0$
- Factor: $(u-4)(u-9) = 0$, so $u = 4, 9$
- Solve: $x^2 = 4$ gives $x = \pm 2$; $x^2 = 9$ gives $x = \pm 3$
- Find $y$: When $x = 2$, $y = 3$; when $x = -2$, $y = -3$; etc.
- Answer: $(2, 3)$, $(-2, -3)$, $(3, 2)$, $(-3, -2)$
Key insight: Symmetric systems often have symmetric solutions.
🔗 Related Topics
- Quadratics — Many nonlinear systems involve quadratics
- Geometry — Systems often represent geometric intersections
- Word Problems — Systems model real-world situations
- Parameter Analysis — Systems with parameters require special techniques
📝 Practice Checklist
- Master 2×2 linear systems
- Practice 3×3 linear systems (AMC12)
- Learn nonlinear system techniques
- Practice geometric applications
- Understand parameter analysis
- Work on speed and accuracy
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