Solving Systems by Addition

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SWBAT solve systems in which adding on both sides, leads to an equation in one variable.

Big Idea

Knowing that the sum of the left and right sides of two different equations are equal, is key to solving certain systems, and later, many real world problems.


15 minutes

Students receive an Entrance Ticket and are asked to answer each question. They should find that the two given expressions are both true statements, the sums of the left and right sides are equal, and that the same holds true with other values. I walk around to see students working and checking to see how they are answering Questions 4 and 5.

The sum of the two equations in the system (3rd equation) is 1.10d + 1.25q = 31. 85 When students substitute the system solution (7,21), into this 3rd equation, they will see that it makes the statement true. I try and get students to explain or make inferences with respect to solving systems of equations. I expect students to write that systems can also be solved by adding or combining both equations. 

I inform the class that today's lesson is about combining the equations in systems to algebraically solve these. This 3rd method of solving systems can be called Linear Combination or Elimination. 


New Info

15 minutes

At this point, students are asking themselves how we are going to do this. After all, the solution to the system they just worked with, was given, (7, 21). How do we use Linear Combination to solve a system we don't know the solution for? I proceed to write a system of equations on the board like the one below, and ask students to try and solve this system without using the substitution method learned in the previous lesson. In other words, I tell the class not to solve the first equation for one of the variables like they do when using the Substitution method. Again i walk around to assess students working.

 x + y = 9

 2x - y = -3

My experience here is that most all of the students combine and quickly see that they are left with an equation in one variable, and proceed to solve for that variable.  (I always try to keep this first system quite simple, because all I want students to see here is that one of the variables must cancel out, so we can solve for the other)

Once students are done, I call on a volunteer to perform the work on the board for all to see and discuss, and end this section by asking, "What do you think the goal of the Addition, or Liner Combination method to solve systems is?" I hope that my students will correctly state the goal is to eliminate one of the two variables so that an equation in only one variable remains, and we could solve it. When a student responds correctly, I say, "Exactly, and do we stop there? Are we done?" I hope students  will remember that we are not done. I want them to say, "No, because we have to find the other coordinate of the solution."



25 minutes

For the application section and corresponding activity, I will ask my students to work in small groups, preferably pairs. I make sure that members of each small group are of similar abilities with respect to their work on systems so far. As I handout the application sheets, Systems by Addition Application, I quietly assign one of the problems of the sheet to seven of my class pairs and tell them that they will be working the indicated problem up on the board. 

I indicate the less complex questions to struggling students and the tougher questions to those students who can probably handle these.  I try to be careful with differentiating here because I don’t want students to pick up the fact that I gave the easier problems to those students the class may see as the “slower” students, and vice versa.  I tell these chosen pairs that they must do the entire worksheet but to try and make sure they get the indicated question right. I obviously walk around and help out to make it work for them.

As I go around assessing students working, I try to get students to do as many of the seven problems as they can, in the first 15 minutes of the activity. I divide the whiteboards up into 7 large sections so that students write their work. The board sections should be large so other students can write feedback or questions near it (see Board divided). Comments work with post-it notes as well. If dividing the boards is a problem, using Chart Paper works great. 

Once students are done, I ask those groups I previously assigned questions to, to go up to the board and write their solutions. They should write the number of the question, the problem, and their answer.

When everyone is done, I ask the class to stand and go around making sure they see the work of all seven questions. To avoid chaos, I line the groups up asking that they go through each problem consecutively from 1 to 7, observe the work, and make a comment or question. If there is no question and they think the work is correct, I encourage students to make a positive remark about it. “Well done”, “nice”,  “correct”, are good examples. Students may often state, “you skipped a step”, or “you forgot something”.  Students should not make corrections, just ask a question, or make a comment.  At the end of the activity, we go through and discuss the questions as a whole group. 


10 minutes

To close the lesson, I walk through each of the board sections, observe the work, and read some of the messages left by the class, pausing where I see a question or comment expressing doubt. I ask the group that worked that particular problem on the board to answer those questions and/or address the comments made. This particular section flows smoothly. Most of the time, the problems on the board are all correctly worked out. Struggling students may have had difficulty with the last two questions, which were word problems. I get more information about this when going over their application sheets and homework assignment.  

Before I collect student work, I ask that they draw a sad, serious, or smiley face (see faces) beside any of the questions on the sheet. Students do this quite quickly, and very little time is wasted. Yet, I get a good idea of how students are feeling with respect to the work. 


The Homework Assignment starts off with straight forward systems to be solved by addition, like the application activity in the lesson, followed by "sum-difference" problems. Question 7 is a favorite of mine. Students may encounter a couple of elements which we did not really see in this lesson, but I expect my students should be able to handle. The first is having to change units of measure. The second is deciding to re-arrange the terms in the equation of a system, properly organizing variables and symbols.


D + M = 149

M - D = 23

Rearranging the second equation gives us...

D + M = 149

-D + M = 23

The answer to this question is that Margo is 7ft 2in. and Debbie is 5ft. 3in.