Precalc
Section sys.eq

Systems of Equations

Systems of equations show up on Exam 4 as a recitation topic from Week 10. They're quick points if you know substitution and elimination — and the nonlinear versions tie directly into the conics you'll see later on the same exam.

⚡ Quick Summary

Imagine two friends each drawing a line on the same piece of graph paper. A system of equations is just asking: "Where do those lines cross?" The crossing point is the answer — an (x,y)(x, y) pair that makes both equations true at the same time. Sometimes lines cross once, sometimes never (parallel lines), and sometimes they're the same line (infinite crossings).

Substitution Method
1.  Solve one equation for y2.  Substitute into the other equation3.  Solve for x4.  Back-substitute to find y\begin{aligned} &1.\;\text{Solve one equation for } y \\ &2.\;\text{Substitute into the other equation} \\ &3.\;\text{Solve for } x \\ &4.\;\text{Back-substitute to find } y \end{aligned}

Best when one equation is already solved for a variable, or a coefficient is 11 or −1-1

Elimination Method
Multiply to match coefficients  ⟹  add/subtract equations  ⟹  solve  ⟹  back-substitute\text{Multiply to match coefficients} \;\Longrightarrow\; \text{add/subtract equations} \;\Longrightarrow\; \text{solve} \;\Longrightarrow\; \text{back-substitute}

Best for linear systems when coefficients line up or nearly line up

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Imagine two friends each drawing a line on the same piece of graph paper. A system of equations is just asking: "Where do those lines cross?" The crossing point is the answer — an (x,y)(x, y) pair that makes both equations true at the same time. Sometimes lines cross once, sometimes never (parallel lines), and sometimes they're the same line (infinite crossings).

You have two main tricks for finding the crossing point. Substitution is like solving a puzzle one piece at a time: if one equation already tells you y=2x−1y = 2x - 1, just swap that into the other equation so you only have xx to worry about. Elimination is more like balancing a scale — you line up the equations and add or subtract them so one variable cancels out completely.

Things get more interesting when the equations involve curves instead of straight lines — like a line meeting a circle or a parabola. These nonlinear systems can cross at more than one point (a line can hit a circle at 0, 1, or 2 spots). The strategy is usually substitution: replace one variable to get a single equation you can solve.

The most important habit: always check your answers in both original equations. Sometimes the algebra creates fake solutions (called extraneous solutions) that look right but don't actually work. And remember — your answer is a point (x,y)(x, y), not just a single number. Don't forget to find both coordinates!

Key Formulas

Substitution Method
1.  Solve one equation for y2.  Substitute into the other equation3.  Solve for x4.  Back-substitute to find y\begin{aligned} &1.\;\text{Solve one equation for } y \\ &2.\;\text{Substitute into the other equation} \\ &3.\;\text{Solve for } x \\ &4.\;\text{Back-substitute to find } y \end{aligned}

Best when one equation is already solved for a variable, or a coefficient is 11 or −1-1

Elimination Method
Multiply to match coefficients  ⟹  add/subtract equations  ⟹  solve  ⟹  back-substitute\text{Multiply to match coefficients} \;\Longrightarrow\; \text{add/subtract equations} \;\Longrightarrow\; \text{solve} \;\Longrightarrow\; \text{back-substitute}

Best for linear systems when coefficients line up or nearly line up

Number of Solutions
Line–Line: 0,1, or ∞Line–Conic: 0,1, or 2Conic–Conic: 0,1,2,3, or 4\text{Line–Line: } 0, 1, \text{ or } \infty \qquad \text{Line–Conic: } 0, 1, \text{ or } 2 \qquad \text{Conic–Conic: } 0, 1, 2, 3, \text{ or } 4

A line can be tangent to a conic (1 solution) or miss it entirely (0 solutions)

Key Takeaways

  • ✓Substitution works best when one equation is already solved for a variable; elimination works best when coefficients line up
  • ✓Nonlinear systems can have 0,1,2,30, 1, 2, 3, or 4+4+ solutions — always expect and check for multiple intersection points
  • ✓The answer is a point (x,y)(x, y), not a single number — always back-substitute to find both variables
  • ✓Check every solution in BOTH original equations — squaring can introduce extraneous solutions

⚠️ Common Mistakes

  • ✗Finding xx but forgetting to find yy — the answer is a point (x,y)(x, y), not a single number; always back-substitute
  • ✗Not checking solutions in BOTH original equations — especially in nonlinear systems, squaring can introduce extraneous solutions
  • ✗Dividing both sides of an equation by a variable (like xx) — you might be dividing by zero and losing a valid solution where x=0x = 0
  • ✗Giving up when you get a negative discriminant — that just means no real intersection points; clearly state "no solution" and explain why
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