# Lesson 16

Solving Systems by Elimination (Part 3)

### Lesson Narrative

This is the last lesson in a series of three lessons on solving systems of equations by elimination. Two new ideas are introduced here.

The first idea is that we can multiply one or both equations in the system by a factor to make it possible to eliminate a variable. Prior to this point, students worked only with systems where at least one variable in the equations had the same coefficient or with opposite coefficients, making the variable removable when the equations were added or subtracted.

Here students see that this is not a requirement for a system to be solvable by elimination. We can first multiply one or both equations by a factor—chosen strategically so that the coefficients of one variable become equal or opposites. Then, the variable can be eliminated by adding an original equation and the new equation, or by subtracting one from the other.

The second new idea is that, whenever we multiply equations in a system by a factor, add or subtract the equations, or otherwise manipulate the equations, we are essentially creating an equivalent system that would help us get closer to finding the solution of the original system.

The work here builds on earlier work on acceptable moves and equivalent equations. Students should understand and be able to explain two key threads:

• Multiplying two equal things by the same value results in two things that are also equal. Variable values that make the original equation true also make the new equation true.
• Adding one equation in a system to another equation is an example of adding an equal amount to each side of an equation. The two sides of the resulting equation are still equal. If the original equations in the system share a set of variable values that make them true, the new equation also shares this set of values.

As they work to process and articulate these key ideas, students practice constructing logical arguments (MP3).

### Learning Goals

Teacher Facing

• Recognize that multiplying an equation by a factor creates an equivalent equation whose graph is the same as that of the original equation.
• Solve systems of equations by multiplying one or both equations by a factor and then adding or subtracting the equations to eliminate a variable.
• Understand that solving a system by elimination or by substitution entails creating one or more equivalent systems that would enable us to solve the original one.

### Student Facing

• Let's find out how multiplying equations by a factor can help us solve systems of linear equations.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Be prepared to display a graph using technology for all to see.

### Student Facing

• I can solve systems of equations by multiplying each side of one or both equations by a factor, then adding or subtracting the equations to eliminate a variable.
• I understand that multiplying each side of an equation by a factor creates an equivalent equation whose graph and solutions are the same as that of the original equation.

Building On

### Glossary Entries

• elimination

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).

• equivalent systems

Two systems are equivalent if they share the exact same solution set.

• 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

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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