Algebra I · 35 minutes
Suggested grades 9–10 · Unit 1: Expressions and equivalent equations · Lesson 2 of 2
Linear Equations and a Parameter
Preserve equality while solving both numerical and letter-coefficient equations.
Mr. S Teaches · Original starter lesson · HS-ALG1-02 · Grade range is planning guidance.Watch, notice, discuss
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Linear equations with unknown coefficients | Mathematics I | High School Math | Khan Academy
By Khan Academy
Work with letter coefficients and compare the solved formula with the lesson’s parameter exception.
Open on YouTube: Linear equations with unknown coefficients | Mathematics I | High School Math | Khan Academy (opens a new tab) ↗Reasoning with linear equations | Solving equations & inequalities | Algebra 1 | Khan Academy
By Khan Academy
Review operations that preserve equivalence; pause to discuss why division by zero is excluded.
Open on YouTube: Reasoning with linear equations | Solving equations & inequalities | Algebra 1 | Khan Academy (opens a new tab) ↗When is dividing by a parameter expression valid, and what happens in the exceptional case?
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Learning objective
Solve and verify a linear equation and state an exception for a parameter formula.
Before you begin
- Distributive property
- Inverse operations
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- parameter
- A value treated as fixed while another variable is solved for.
- solution set
- All values making an equation true.
- identity
- An equality true for every allowed value of a variable.
Understand the concept
Equivalent equation steps preserve the same solutions when the operation is reversible. Dividing by an unknown coefficient requires a condition that the coefficient is not zero. Letters are not permission to ignore exceptional cases. A simplified true statement indicates every input works; a false statement indicates no solution.
Teach it step by step
- Expand 3(2x−1)=4x+9, collect x terms and solve while naming each equality-preserving operation.
- Check the candidate in the original equation, not only the last transformed line.
- Solve ax+b=c for x when a≠0 and record that condition beside the formula.
- Handle a=0 separately: decide whether b=c or b≠c, then describe the solution set.
Make the thinking visible
Worked example or model response
6x−3=4x+9 gives 2x=12 and x=6. Original check: 3(12−1)=33 and 24+9=33. For ax+b=c with a≠0, x=(c−b)/a. If a=0 and b=c all x work; otherwise none do.
Guided practice
Solve 5(x+2)=3x+18, then solve mx=12 and discuss m=0.
Independent practice
1. Solve 4−2(x−3)=18. 2. Solve px+7=7 for p≠0, then discuss p=0.
Quick assessment
Why is dividing by a−a never a valid cancellation step?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: 2x=8, x=4. For mx=12, x=12/m if m≠0; no solution when m=0.
- Independent 1: x=−4; substitution gives 18. 2: x=0 if p≠0; every x works if p=0.
- Quick check: a−a is zero, and division by zero is undefined.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Annotate one operation per line and test exceptional cases with numbers before letters.
Extension challenge
Create one parameter equation whose solution changes from unique to none at a specified parameter.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSA.REI.B.3
Common Core · High school · Algebra
Regular high school mathematics standard
Solve linear equations, including coefficients represented by letters.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.