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Grade 8 · Mathematics · 30 minutes

Unit 2 · Functions and linear systems · Lesson 2 of 2

Systems for Two Ticket Types

Use two constraints to solve for two unknown ticket counts.

Mr. S Teaches · Original starter mini-lesson · MS8-MATH-03

Learning objective

Solve a two-equation system and explain how the solution satisfies both constraints.

Before you begin

  • Two-step equations
  • Substitution
  • Coordinate graphing

Gather your materials

  • Paper and pencil
  • Ruler and graph paper

Language to know

system
Equations considered together with shared variables.
intersection
A point belonging to both graphs.
simultaneous solution
Values that make every equation in a system true.

Understand the concept

One equation with two unknowns may allow many pairs. A second independent constraint narrows the possibilities. A system’s solution is a pair, not just one of its numbers. Substitution replaces a variable with an equal expression. On a graph, a shared solution is a point on both lines; parallel distinct lines have no shared solution.

Teach it step by step

  1. Define a and s as the counts of adult and student tickets. Use 10 tickets total and prices $8 and $5 to write a + s = 10 and 8a + 5s = 62.
  2. Replace s with 10 − a in the money equation; distribute and collect like terms.
  3. Solve 3a = 12, then use the total-count equation to find s.
  4. Check both original constraints including units, then plot the lines s = 10 − a and s = 12.4 − 1.6a.
  5. Contrast y = 2x + 1 and y = 2x + 4 and explain why their equal slopes give no intersection.

Make the thinking visible

Worked example or model response

8a + 5(10 − a) = 62 → 3a + 50 = 62 → a = 4. Then s = 6. The checks are 4 + 6 = 10 tickets and 8(4) + 5(6) = $62. The graphs meet at (4,6).

Guided practice

Solve x + y = 9 and 2x + y = 13 by subtracting the first equation from the second. Check the ordered pair.

Independent practice

1. Solve x + y = 12 and 3x + 2y = 29. 2. Explain whether y = x − 2 and y = x + 3 can share a solution.

Quick assessment

Why is a pair that satisfies only the total-ticket equation not necessarily a solution to the ticket problem?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: x = 4, y = 5; both equations hold.
  2. Independent 1: x = 5, y = 7; 5+7=12 and 15+14=29. 2: No; the two lines have equal slopes and different intercepts.
  3. Quick check: It must also satisfy the money constraint; meeting only one condition is insufficient.

Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.

Make it work for your learner

More support

List possible whole-number pairs for the total count before substituting algebraically.

Extension challenge

Create a system with infinitely many solutions and explain how one equation repeats the other.

Selected contributions · Official sources

Standards connections

Contributes to Common Core standard

8.EE.C.8.a

Common Core · Grade 8

Interpret the shared solution as a graph intersection.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.

Contributes to Common Core standard

8.EE.C.8.c

Common Core · Grade 8

Solve a contextual problem represented by two linear equations.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.