Algebra I · 35 minutes
Suggested grades 9–10 · Unit 2: Linear models and simultaneous constraints · Lesson 1 of 2
Two Constraints at a Market Stall
Use a two-equation system to distinguish total items from total revenue.
Mr. S Teaches · Original starter lesson · HS-ALG1-03 · Grade range is planning guidance.Watch, notice, discuss
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The substitution method | Systems of equations | 8th grade | Khan Academy
By Khan Academy
A prerequisite refresher on substitution before using the method in the market-stall context.
Open on YouTube: The substitution method | Systems of equations | 8th grade | Khan Academy (opens a new tab) ↗Systems of equations with elimination (and manipulation) | High School Math | Khan Academy
By Khan Academy
Compare elimination with substitution, then check the resulting quantities against both market constraints.
Open on YouTube: Systems of equations with elimination (and manipulation) | High School Math | Khan Academy (opens a new tab) ↗Solving systems of linear equations with substitution example | Algebra II | Khan Academy
By Khan Academy
Use another substitution example to discuss why the solution must satisfy both equations.
Open on YouTube: Solving systems of linear equations with substitution example | Algebra II | Khan Academy (opens a new tab) ↗How can one ordered pair satisfy both constraints, and how will you verify it?
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Learning objective
Solve a contextual linear system and verify both constraints.
Before you begin
- Linear equations
- Substitution or elimination
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- system
- Equations with shared variables considered together.
- elimination
- Combining equations to remove one variable.
- intersection
- A point lying on both graphs.
Understand the concept
A stall sells small bags for $3 and large bags for $5. Item count and revenue describe different constraints. A pair of values solves the problem only when it satisfies both. Elimination works because equal combinations of equal quantities remain equal; interpreting the pair prevents an algebraic result from becoming an unlabeled answer.
Teach it step by step
- Define s and l, then translate 18 total bags and $70 revenue into s+l=18 and 3s+5l=70.
- Multiply the first equation by 3 and subtract it from the second.
- Solve for l, find s and check both original statements.
- Sketch the two lines with l on the vertical axis and relate their intersection to the pair.
Make the thinking visible
Worked example or model response
3s+3l=54; subtracting gives 2l=16, so l=8 and s=10. Checks: 10+8=18 bags and 30+40=$70. The graphs intersect at (s,l)=(10,8).
Guided practice
Solve x+y=11 and 2x+5y=37 by eliminating x. Explain the ordered pair.
Independent practice
1. Solve 2x+y=9 and x−y=3. 2. Explain why x+y=4 and 2x+2y=10 cannot both hold.
Quick assessment
Is satisfying the revenue equation alone enough to solve the stall problem?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: 2x+2y=22; 3y=15, y=5, x=6.
- Independent 1: x=4, y=1. 2: Doubling the first gives 2x+2y=8, incompatible with 10.
- Quick check: No. The pair must also satisfy the total-item count and any contextual restrictions.
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
List a few count pairs and corresponding revenues before using elimination.
Extension challenge
Change the revenue so that the equations have a fractional solution and explain why that cannot represent whole bags.
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.C.6
Common Core · High school · Algebra
Regular high school mathematics standard
Solve and graph a pair of linear equations and interpret their shared solution.
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.