Lesson 2: Number of Solutions

How can you tell if an equation will have no solution just by looking at the variable terms and constants?What does it mean for a value to be a "solution" if both sides of the equation are identical (a=a)?

Identity (Infinitely many solutions)

Contradiction (No solution)

Linear Equation

8.EE.C.7.a: Give examples of linear equations with one solution, infinitely many solutions, or no solutions. It requires showing these possibilities by transforming equations into simpler forms until they result in x = a, a = a, or a = b.
8.EE.C.7.b: Solve linear equations with rational number coefficients (including fractions and decimals). This includes equations that require expanding expressions using the distributive property and collecting like terms.

Target D.
Analyze and solve linear equations and pairs of simultaneous linear equations.
The student identifies and writes examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.

Description: Students will explore the three possible outcomes of solving linear equations.

Purpose: To help students move beyond simple computation and begin analyzing the algebraic structure of equations.

DOK Level: 2 (Skill/Concept)

The Best Deal Comparison (One Solution):
Compare two different cell phone plans or streaming services where one has a higher monthly fee but a lower data cost. Students can calculate the exact point where the total costs are identical, representing the single solution where both plans are equally beneficial.

Parallel Pay Rates (No Solution): Discuss a situation where two people have the same hourly commission rate but different base salaries (e.g., Job A pays $15/hr + $50 bonus while Job B pays $15/hr + $20 bonus). Students can analyze why, no matter how many hours are worked, the total earnings will never be the same, illustrating no solution.

Unit Conversion or Currency Exchange (Infinitely Many Solutions):
Use two different but equivalent mathematical formulas to describe the same real-world relationship, such as different ways to write the formula for converting Celsius to Fahrenheit. Since the two expressions represent the exact same rule, every input will result in equality, demonstrating infinitely many solutions.

The "No Solution" Confusion: Students may think they made a calculation error when the variables cancel out and leave a false statement (e.g., 5 = 2), rather than recognizing it as a no-solution case.

Infinitely Many vs. Zero:
Students often confuse an equation that results in 0 = 0 (infinitely many solutions) with an equation where the solution is actually the number zero (x = 0).

Misinterpreting Identical Coefficients:
Students might assume that any equation with the same variable terms on both sides automatically has infinite solutions, overlooking the importance of the constant terms.

Visual Anchor Charts:
Provide a desk reference or anchor chart that explicitly shows the three outcomes (x = a, a = a, a = b) with simple visual examples for each.

Card Sort Activity:
Use a "hands-on" card sort where students categorize various equations into "One," "None," or "Infinite" columns based on their algebraic structure before solving.

Step-by-Step Flowchart:
Offer a flowchart for students who struggle with multi-step processing, guiding them through the distributive property, combining like terms, and finally comparing the resulting sides.

-Exit Tickets

-Quiz

-Check-Ins

-Performance Task

Manipulatives

Worksheets

Google Forms

Textbook