This is the first of three lessons that develop the idea of solving systems of linear equations in two variables by elimination.

Students examine a diagram of three hangers where the third hanger contains the combined contents of the first two hangers and all three hangers are balanced. Then, they analyze the result of adding two linear equations in standard form and notice that doing so eliminates one of the variables, enabling them to solve for the other variable and, consequently, to solve the system. In studying and testing a new strategy of adding equations and then offering their analyses, students construct viable arguments and critique the reasoning of others (MP3).

Next, students connect the solution they found using this method to the graphs of the equations in a system and the graph of the third equation (that results from adding or subtracting the original equations). They observe that the solution they found is the solution to the system, and that the graph of the third equation intersects the other two graphs at the exact same point—at the intersection of the first two.

The foundational idea is that adding or subtracting equations in a system creates a new equation whose solutions coincide with those of the original system. Students begin using this insight to solve systems, but they are not yet expected to construct an argument as to why this approach works.