Unit 08 · Lesson 02 · Open-Age

Solving Nonlinear Systems by Substitution

Substitution reduces a nonlinear system to one equation. Each resulting input must be paired with its output and checked.

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Learning objectives

Explain the central structure of solving nonlinear systems by substitution using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.

Essential question: How can multiple equations, matrices, and graphical intersections describe systems with several interacting constraints?

Governing relationship

Substitute one expression into the other equation.

Substitution reduces a nonlinear system to one equation. Each resulting input must be paired with its output and checked.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

x+y=5, y=x²−1.Substitute: x+x²−1=5 → x²+x−6=0 → x=2,−3; points (2,3),(−3,8).

y=√x and y=x−2.Solve √x=x−2 with domain x≥2, then check candidates.

Guided and independent practice

Solve a line–quadratic system.
Solve a radical system with checks.
Interpret multiple intersections.
Create one original example involving solving nonlinear systems by substitution and solve it completely.
Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.

Differentiated pathways

Foundation

Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.

Core

Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.

Extended

Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.

Application laboratory

Use substitution to find break-even or equilibrium points between nonlinear models.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Forgetting that each x-solution creates a full ordered pair.

Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.

Mastery evidence

Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.

Reflection

What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?