BetterGrades Precalculus · Unit 8 · Lesson

Nonlinear systems

Solve systems involving lines, quadratics, circles, rational, exponential, or other nonlinear functions and interpret all real intersections.

Opening

Start with the situation

A nonlinear system contains at least one curved relation and may have zero, one, or several intersections.

Systems combine several conditions into one decision. Graphs, equations, inequalities, and matrices are different views of the same requirement: the final result must satisfy every condition at once.

Before you begin

Prerequisite check

Solve equations and systems.
Interpret graphs as solution sets.
Use organized arithmetic and units.

Core explanation

Explanation

Substitute an isolated equation, solve the resulting one-variable equation, recover the other coordinate, and check every candidate.

A repeated root often represents tangency; a negative discriminant can show no real line-parabola intersections.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through nonlinear intersection gallery, substitution workflow, or another equivalent representation.

Conceptual reading

What the idea is really doing

A system asks for simultaneous truth. Graphs show common intersections, elimination preserves the solution set, and matrices record the same operations compactly; the representation changes, but the solution condition does not.

This lesson narrows that lens to one goal: solve systems involving lines, quadratics, circles, rational, exponential, or other nonlinear functions and interpret all real intersections. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Substitute an isolated equation.
Solve the resulting one-variable equation.
Recover the other coordinate.
Check every candidate.

Verification: Substitute the result into every original equation or inequality. For matrix work, translate the final rows back into statements about variables, pivots, free variables, and consistency.

Foundation walkthrough

Plan before calculating

Problem

Solve $y=x^2 \qquad y=2x-1$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Substitute an isolated equation, solve the resulting one-variable equation, recover the other coordinate, and check every candidate.
Conclusion: $(1,1)$
Why the check works: The repeated root represents tangency.

Worked examples

See the idea in three forms

foundation example

Solve $y=x^2 \qquad y=2x-1$

Solution $(1,1)$

The repeated root represents tangency.

representation example

Solve $y=x^2 \qquad y=-1$

SolutionNo real solution.

This example expresses nonlinear systems in a second form.

transfer example

Meaning double root.

SolutionTangency.

A repeated root often represents tangency; a negative discriminant can show no real line-parabola intersections.

Nonlinear intersection gallery. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The repeated root represents tangency. — **Anchor figure · Nonlinear intersection gallery**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The repeated root represents tangency.

Substitution workflow. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for nonlinear systems. — **Mechanism figure · Substitution workflow**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for nonlinear systems.

Repeated root as tangency. Compare the valid path with the tempting shortcut. The figure shows why listing only x-values instead of complete intersection points leads to a false conclusion. — **Comparison and error figure · Repeated root as tangency**
Compare the valid path with the tempting shortcut. The figure shows why listing only x-values instead of complete intersection points leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is listing only x-values instead of complete intersection points.

Check yourself

Solve $y=x^2 \qquad y=4$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Solve $y=x^2 \qquad y=4$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Solve $y=x^2 \qquad y=-1$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Meaning double root.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Why graph first?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: $(1,1)$ . Use the foundation problem as evidence: Solve $y=x^2$ and $y=2x-1$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of nonlinear systems that it illustrates: Solve $y=x^2 \qquad y=-1$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is listing only x-values instead of complete intersection points.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Solve $y=x^2$ and $y=2x-1$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: Meaning double root. Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for nonlinear systems, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Systems of inequalities and feasible regions, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Solvey=x2y=2x−1y=x^2 \qquad y=2x-1y=x2y=2x−1

Solvey=x2y=2x−1y=x^2 \qquad y=2x-1y=x2y=2x−1

Solvey=x2y=−1y=x^2 \qquad y=-1y=x2y=−1

Meaning double root.

Original BetterGrades manuscript, rights-separated references.

Solve $y=x^2 \qquad y=2x-1$

Solve $y=x^2 \qquad y=2x-1$

Solve $y=x^2 \qquad y=-1$