This is the first of three lessons that develop the idea of solving systems of linear equations in two variables by elimination.
Students warm up to the idea of adding equations visually. They 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.
- Recognize that adding or subtracting equations in a system creates a new equation with a solution that coincides with that of the original system, so the new equation can be used to solve the original system.
- Solve systems of equations by adding or subtracting the equations strategically to eliminate a variable.
- Use graphing technology to graph the sums and differences of the equations in a system, and analyze and describe (orally and in writing) the behaviors of the graphs.
- Let’s investigate how adding or subtracting equations can help us solve systems of linear equations.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (If students typically access the digital version of the materials, Desmos is always available under Math Tools.)
- I can solve systems of equations by adding or subtracting them to eliminate a variable.
- I know that adding or subtracting equations in a system creates a new equation, where one of the solutions to this equation is the solution to the system.
A method of solving a system of two equations in two variables where you add or subtract a multiple of one equation to another in order to get an equation with only one of the variables (thus eliminating the other variable).
solution to a system of equations
A coordinate pair that makes both equations in the system true.
On the graph shown of the equations in a system, the solution is the point where the graphs intersect.
Substitution is replacing a variable with an expression it is equal to.
system of equations
Two or more equations that represent the constraints in the same situation form a system of equations.
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