On Substitution (A continuation of "Catching Up")

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SWBAT solve systems using substitution when equations are in different forms.

Big Idea

Group activity, Land, or Money problems can be solved using Systems. Many lend themselves to substitution.


15 minutes

The Launch Question can be handed to students as an entrance slip, or it could be projected on the whiteboard and students asked to write the numbers 1 to 4 in the order they think they belong. Students must number the systems in increasing order of difficulty to solve by substitution. In other words, number one should be the system that most lends itself to solving by substitution, and (4), the hardest to solve by substitution.

I plan to allow my students to chat about it with their elbow partners. Before letting them chat, I'll let them know that I'm going to call on someone to answer and explain the criteria he/she used to order the systems.

In my view, the expected order is as follows:

            (  3   )

x + y + 10

2x + 3y = 6


            (   4   )

 2x + 7y = 10

3x - 4y = -12

        (   1  )

y = 2x

y = 3x + 5          

       (  2   )

 y = -3x

5x - y = 20

The criteria I expect to hear from students are:

  1. The third system is the easiest to solve by substitution because both equations are solved for y. 
  2. The next easiest system to solve by substitution is the last one where one of the equations is solved for y. 
  3. The first system has variables in its equations with a coefficient of 1, making it easy to solve for any of the variables, and then substitute. 
  4. The second system in appearance is the hardest to solve by substitution because neither of the equations is solved for a variable, and no variable has a coefficient of 1, making it more difficult, because solving for x, or for y in either of the two equations would give us a fraction; the system can be solved this way, but it just makes things a bit more complex. 


35 minutes

For this section, the class should be divided in small groups, each group with an application problems handout: Application section On Substitution.

In each of the problems on the handout, one or both of the equations in the system begin in the linear combination form Ax +By = C. However, in each system, in one or both of the equations, it is easy to solve for one of the variables. Then, substitution can be used to solve it like in the previous lesson.

In Problem 1, to help struggling students come up with the equations, I point out that one equation should refer to the amount of coins Grandpa has.

Let d = the number of dimes, and q = the number of quarters, leading them to writing the first equation.

I also indicate that the second equation of the system represents the total value of the coins.

How much is each dime worth? (0.10) How much is each quarter worth? (0.25).

I try leading the class towards successfully writing the second equation without giving it away. Frequently, it's the writing of the equations to model a situation that troubles my students, despite having done this type of work numerous times before. As we handle a few different types of problems, I try to help students learn the similarities among the situations so that they can reason by analogy.

Once students solve for one variable, I find that they sometimes forget to continue solving. I remind the students not to lose sight of the fact that they are searching for a point that relates two values.  When substituting back into the equations to find the other variable, I try to help them make the correct judgment when substituting back. By correct, I mean substituting into the easier of the two. I don't simply tell them which one to use, however, instead I let them decide. They often figure out that there was an easier approach as they work. 

For Problem 2, I tell students that it is quite similar to the first problem. One equation represents the amount of land, and the second represents the total cost of the land. After doing Problem 1, students usually have little difficulty with writing the equations and solving this problem. 

Students note that the third system is slightly different. I may indicate that one of the equations is already solved for one of the variables. In this problem, just like in the previous ones, I point out that before translating the situation into an equation, they should identify the two variables involved. In this case, the first equation could be " b= 2c" because they sold twice as many brownies than cookies. The second equation just like the second equations in the previous problems, refers to the value, or cost, of the items involved.

I make sure that students show all work on the worksheet, and call on members of 3 different groups to go up to the board and write their work. Once the work is on the board I ask the class if any of the systems could be solved differently. What I want students to see is that in either of the first two problems, one of the equations can be solved for x, as well as for y, and that either way obviously leads to the same solution point. I always have someone actually go up and demonstrate this. 


10 minutes

To bring the lesson to a close I tell the class that in the last 10 minutes I want them to draw faces on their application worksheets. I tell students to draw faces in corresponding areas of their work as follows:

A "happy face"= I really understood this idea or procedure

A "serious face" = I still have questions before I can say I understand

A "sad face" = I am very confused here..

I also tell the class to add any short comment if they feel it is necessary. 

I collect the student's work before they leave, check some of the faces, and go over as many serious and sad faces as time permits. After class, I go through all of the work and check for common places of confusion or doubt so I can address these during our next class, or during after school coaching. 


Students should be able to do the Homework on Substitution problems smoothly. I ask that they show all work as always and that they highlight the solution to each system. My homework assignment would be to check the smiley faces and student classwork and decide what to address in my next class in order to improve mastery of the lesson objectives. 


The two extension tasks can be given to more advanced students, yet eventually, to all students. In the first system, solving for one of the variables leads to working with fractions, yet these systems will be solved by linear combination or elimination in lessons that follow. 

In the second task, students are asked to create a system of equations using the students in their own classroom.

Example: There are 25 students and there are 6 more students wearing jeans than not wearing jeans, etc.

Then, students must solve the system to their problem. 

 See the EXTENSION On Substitution for more details.