Lesson 1: Systems of Linear Equations

If two lines have the same slope, but different y-intercepts, how many times do they intersect?

If two lines have different slopes, how many times do they intersect?

If two lines have the same slope and same intercept, how many times do they intersect?

system of equations
consistent
independent
dependent
inconsistent

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Lesson Description This lesson defines what a system of equations is and explores the visual and algebraic meaning of a solution. Key concepts include: The Definition of a System: Understanding that a system consists of two or more equations with the same set of variables. Finding the Solution by Graphing: Plotting two lines on a single coordinate plane and identifying the point of intersection (x, y). Verifying Solutions: Checking a potential solution by substituting the (x, y) values into both equations to ensure they result in true statements. Types of Systems: Categorizing the three possible outcomes of a system: Consistent and Independent: The lines intersect at exactly one point. Inconsistent: The lines are parallel and have no solution. Dependent: The lines are identical and have infinitely many solutions.
Purpose
The purpose of Section 12.1 is to develop multi-variable logic. Most real-world problems involve more than one constraint (e.g., "I have $50 to spend AND I need to buy exactly 10 items"). This section teaches students how to find the "sweet spot" where multiple conditions are met simultaneously. It serves as the foundational logic for business "break-even" analysis and is the gateway to the more advanced algebraic methods (substitution and addition) covered later in the chapter.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): Students must perform a coordinated set of actions: graphing two distinct functions and then interpreting their geometric relationship. It requires identifying a specific shared coordinate and verifying it through substitution. Students must also categorize the system based on its graphical behavior, which requires comparing the slopes and intercepts of both lines.

Have the students work in pairs and supply them with a series of linear equations, have them pick any two equations and see how many solutions they will have?

doing the "you try" and the practice problems on page 285-288