Lesson 3: The addition(elimination) method

What is an additive inverse? What is the additive inverse of 5y?

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.REI.6 Solve systems of linear equations exactly, focusing on pairs of linear equations in two variables.

Lesson Description This lesson focuses on the power of "additive identity" to simplify complex systems. Students learn to manipulate equations so that adding them together causes one variable to cancel out entirely. Key steps include: Alignment: Ensuring both equations are in General Form so that like terms (x, y, and constants) are vertically aligned. Creating Opposites: Using the Multiplication Property of Equality to create "additive opposites" (e.g., +2y and -2y) for one of the variables. The Addition Step: Adding the two equations together to create a new equation with only one variable. Solving and Back-Substituting: Solving for the remaining variable and substituting the result into either original equation to find the second coordinate. Recognizing Special Cases: Identifying when both variables are eliminated, resulting in a false statement (No Solution) or a true identity (Infinite Solutions).
Purpose
The purpose of Section 12.3 is to promote mathematical efficiency and strategy. While the Substitution Method works for any system, the Addition Method is the "power tool" for systems with larger coefficients or those lacking a variable with a coefficient of 1. Mastering this method reinforces the idea that you can perform operations on entire equations as single units. This logic is a direct prerequisite for understanding Matrix Algebra and Gaussian Elimination in more advanced mathematics and computer science courses.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): Students must demonstrate strategic thinking by deciding which variable is easiest to eliminate and what "multiplier" is needed to create opposites. It requires high-level coordination of integer operations, the distributive property, and multi-step equation solving. The student must "look ahead" to see how their choice of multiplier will affect the rest of the problem.

Kinesthetic Learners: Students may benefit from using concrete models to solve systems of equations with elimination. Have students write the terms of the equations on pieces of paper or use algebra tiles or other models to represent the equations. When they eliminate a variable, have them remove the model for that variable. The act of removing the terms should help them remember eliminating the variable.

Visual Learners: Have students find the missing value. The values represent the sums of each row and column. Hint: Let each symbol or set of symbols represent one variable.

Extension: Write two equations on the board with the constants missing. Have students find the missing constants that ensure the given solution. For example, write the system 3x + 2y = ?, 5x - 2y = ? and tell students that the solution is (1.5, 0.25).

"you try" on page 282 and practice problems 289-291