Unit 2, Lesson 15
Solving Systems by Elimination (Part 2)
Algebra 1
15.1
Warm-up
Standards Alignment
Building On
Addressing
HSF-BF.A.1.b
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Building Toward
HSA-REI.C.5
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students reason about whether and when the sums of equations are true. The work here prepares students for the next activity, in which they begin to think about why the values that simultaneously satisfy two equations in a system also satisfy the equation that is a sum of those two equations.
Launch
NoneActivity
NoneHere is a true equation: .
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Perform each of the following operations and answer these questions: What does each resulting equation look like? Is it still a true equation?
- Add 12 to each side of the equation.
- Add to the left side of the equation and 12 to the right side.
- Add the equation to the equation .
- Write a new equation that, when added to , gives a sum that is also a true equation.
- Write a new equation that, when added to , gives a sum that is a false equation.
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share their responses to the first three questions. Discuss questions such as:
- "Why do you think the resulting equations remain true even after we add numbers or expressions that look different to each side?" (The numbers being added to the two sides are always equal amounts, even though they are written in a different form.)
- "Suppose we subtract 12 from the left side of and subtract from the right side. Would the resulting equation still be true?" (Yes. is 39 and is also 39. The same amount, 12, is subtracted from both sides.)
- "Here is another equation: . Suppose we subtract 12 from the left side of that equation and from the right side. Would the resulting equation be true? Why or why not?" (No. The given equation is a false statement. Subtracting the same amount from both sides will keep it false.)
Invite students to share their equations for the last two questions. Display the equations for all to see. If no one shares an equation that uses a variable, give an example or two (as shown in the Student Responses).
Make sure students understand that adding (or subtracting) the same amount to (or from) each side of a true equation keeps the two sides equal, resulting in an equation that is also true. Adding (or subtracting) different amounts to (or from) each side of a true equation, however, makes the two sides unequal and thus produces a false equation.
15.2
Activity
Materials
To Gather
Graphing technologyActivity Narrative
This activity serves two goals. The first goal is to further build students' intuition about the new variable equation that comes from adding two variable equations in a system, and about the solution to that new equation. This is done by grounding the addition and the sum in a familiar context.
The second goal is to support students in reasoning about why the new equation shares a solution with the original system. The context gives students a concrete mental reference, which can be helpful for interpreting the intersection of the graphs of all three equations and for thinking about the solution that all three equations share.
In this activity, students also encounter a system that they can solve by graphing and by substitution, but that cannot be easily solved simply by adding or subtracting the equations. This observation may pique students' curiosity and make them wonder if it is possible to solve all systems by elimination.
As students work, identify students who solve the system by graphing and those who solve by substitution. Ask them to share their work later.
This activity uses the Three Reads math language routine to support reading comprehension and sense-making.
15.3
Activity
Instructional Routines
NoneMaterials
To Gather
Graphing technologyActivity Narrative
In this activity, students practice using algebra to solve systems of linear equations in two variables and checking their solutions. Students do not have to use elimination, but the equations in the first three systems conveniently have opposites for the coefficients of one variable, so one variable can be easily eliminated.
In the last system, none of the coefficients of the variables are opposites. Some students may choose to solve the system by substitution, but the process would be pretty cumbersome. The complication that students encounter here motivates the need for another move, which students will explore in the next lesson.
Students who opt to use technology to check their solutions practice choosing tools strategically (MP5).
Launch
Keep students in groups of 2, and provide continued access to graphing technology, in case it is needed for checking solutions.
If time is limited, ask one partner in each group to solve the first two of the systems and the other partner to solve the last two, and then ask them to check each other's solutions.
Lesson Synthesis
To highlight the idea that adding equal amounts to the two sides of a true equation keeps the equation true, display this system for all to see:
Ask students if there is a pair of - and -values that makes both equations true. Once students suggest as the solution, use graphing technology to graph the equations and verify the solution. Display the graph for all to see.
Survey the class to see who have solved the system by adding the left side of each equation ( and ) and then adding the right side (7 and 20). (Most or all students are likely to have done this.) Then, discuss with students:
- "Why is it OK to add an expression, , to one side of the first equation, but add a number, 7, to the other side?" (Because and 7 are equal)
- "What is the new equation? () Does the pair still make this equation true?" (Yes. Graph this equation to illustrate that it intersects the original graphs at the same point.)
- "Suppose we add 7 to the left side of the first equation and to the right side. What is the new equation?" () "Would the same pair of and still work?" (Yes. Display the graph of the new equation so students could see that the new line intersects at the same point. Students could also verify by substitution.)
- "Suppose that, instead of adding, we subtract 7 from the left side and subtract from the right side. What is the new equation?" (). "Would the same pair of and still work?" (Yes. Graph the new equation to show that this line, too, intersects at the same point. Or students could verify by substitution.)
- "Why does the new equation still give the same solution even though we didn't add the numbers or expressions from the same side of each equation?" (If the two sides of an equation are equal, it doesn't matter which side of one equation is added to which side of another equation. The new equation will stay balanced.)
Student Lesson Summary
When solving a system with two equations, why is it acceptable to add the two equations, or to subtract one equation from the other?
Remember that an equation is a statement that says two things are equal. For example, the equation says a number has the same value as another number . The equation says that has the same value as 12.
If and are true statements, then adding to and adding to means adding the same amount to each side of . The result, , is also a true statement.
As long as we add an equal amount to each side of a true equation, the two sides of the resulting equation will remain equal.
We can reason the same way about adding variable equations in a system like this:
In each equation, if is a solution, the expression on the left of the equal sign and the number on the right are equal. Because is equal to -1:
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Adding to and adding -1 to 17 means adding an equal amount to each side of . The two sides of the new equation, , stay equal.
The - and -values that make the original equations true also make this equation true.
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Subtracting from and subtracting -1 from 17 means subtracting an equal amount from each side of . The two sides of the new equation, , stay equal.
The -variable is eliminated, but the -value that makes both the original equations true also makes this equation true.
From , we know that . Because 6 is also the -value that makes the original equations true, we can substitute it into one of the equations and find the -value.
The solution to the system is , or the point on the graphs representing the system. If we substitute 6 and 11 for and in any of the equations, we will find true equations. (Try it!)