Lesson 2: The Substitution Method

How can you check a solution of a system of equations?

If the result of solving by substitution is an identity, then how many solutions are there?

If the result of solving by substitution results in a false statement, then how many solutions are there?

substitution

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 focuses on the algebraic process of "replacing" one variable with an equivalent expression. Students will master a systematic four-step procedure: Isolate: Solving one of the equations for either x or y (choosing the variable with a coefficient of 1 or -1 for efficiency).Substitute: Plugging the resulting expression into the other equation, effectively turning a two-variable system into a single-variable equation. Solve: Using standard linear solving techniques to find the value of the first variable. Back-Substitute: Using that value to find the second variable and writing the final answer as an ordered pair $(x, y)$.Special Cases: Recognizing algebraic outcomes for parallel lines (e.g., 5 = 9, which means No Solution) and identical lines (e.g., 0 = 0, which means Infinite Solutions).
Purpose
The purpose of Section 12.2 is to move students toward algebraic precision. Graphing is limited by the scale of the paper and the human eye; if the solution to a system is (2.35, -4.1), a graph will only provide an estimate. Substitution provides an exact answer. This method also reinforces the fundamental algebraic concept of equivalence—the idea that if y = 2x, then y and 2x are interchangeable. This is a vital skill for success in STEM fields and higher-level courses like College Algebra.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section requires a high degree of procedural fluency. Students must follow a multi-step algorithm where an error in the first step propagates through the entire problem. They must make strategic decisions about which variable is the easiest to isolate and carefully manage distributive property and sign changes when substituting expressions into parentheses.

Beginning Ask questions about the lesson content to elicit yes/no answers: “Look at Example 1. Is one of the equations solved for one of the variables?” yes “Is the first step of solving the system complete?” yes

Intermediate/Advanced Ask questions about the lesson content to elicit short answers: “Look at Example 1. Which equation is solved for a variable?” the first “What should be substituted for y in the second equation?” 2x + 1

Advanced High Ask questions about lesson content to elicit complete sentences: “How does Example 1 compare to Example 2?” In Example 1, Step 1 is already completed. “How would you choose the variable to solve for to solve Example 2 by substitution?” Solving the first equation for x makes sense because its coefficient is 1.

"you try" on page 280 and practice problems on page 288-289