Systems and Parallel Lines

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SWBAT determine whether a system has 0, 1, or infinitely many solutions.

Big Idea

Systems having 0, 1, or infinitely many solutions correspond to lines not intersecting, intersecting once, or being coincident.


20 minutes

To begin the lesson I hand each student an Entrance Slip. I ask that they complete the slip together with their immediate partner. Each student will have their own graphing calculator, but should only use them when instructed to do so in the entrance slip. 

Students should make inferences as to when a system of equations results in two lines intersecting, being parallel, or coinciding. They should state that when combining the equations gives a false statement, the lines are parallel and no solution exists. They should also state that if the combination of both equations gives a "duplication" like 16=16, then the lines coincide and the system has infinite amount of solutions. 


1. It may not be clear to students that a "duplication" like 16 = 16 has solutions. I would ask that they think of that sentence as 0x + 16 = 16, and ask which values for x would work here. They will see that any value works. 

2. It's a good idea to give students an example of a system that when combined gives a solution such as x = 0, and show them that this is a system with intersecting lines that intersect where x = 0. Students may think that when they get 0, this means that there is no solution or infinite solutions. I would also ask that they substitute 0 for x to find y, and check using both equations. 



New Info

10 minutes

Each pair of students will receive a New Info Table to complete at their desks. I inform my students that they can use their calculators if they wish and that discussion is encouraged. I usually give information in the New Info section of a lesson, but this time I thought it would be a good idea if the students write in the information themselves. As usual, they should keep this organizer in their notebooks. 

I project the table on the board and once students are finished I ask volunteers to go up to the board to complete the table by rows. 



20 minutes

Today's Application will be worked on in small groups. I like to change pairings  for this activity by asking students to work with their other elbow partner.

Each small group should complete the Application Work Systems and Parallel Lines together. I stroll through the class as students work and monitor their progress. For the first two parts, I ask struggling students to actually write a system of equations to help them answer the questions. Some go ahead and solve the pizza problem to find that a solution exisits and the situation could have occurred. 

In Part III, I make sure students do not graph the equations with their calculators. In Part IV, I ask students to summarize how they can use ratios to find if the system has one, none, or infinite solutions, and write these clues in their notebooks, as if they were writing them to an absent student. I collect their work to assess them and return these following day. 


10 minutes

To end the lesson I write the equation y = 2x + 1 on a side of the board, and I open the graphing calculator on the SmartBoard for all to visualize.

I then make three columns and write the headings:

  • Parallel Lines
  • Coincident Line
  • Intersecting Lines 

Once the headings are written, I call on students to state equations of lines that are parallel to y = 2x + 1 and write systems under that heading. (My students love to volunteer to write the equations on the board as well as to graph them on the Desmos calculator.) As students name equations that are parallel, the lines should be projected on Desmos so that the whole class can see that the lines are parallel.

Next, I ask students for equations of lines that coincide with y=2x+1. (The previous parallel lines should be erased from the desmos calculator for clarity) Students will see if the lines graphed are identical to y=2x+1.

Finally, I ask students for equations of lines that will intersect with the original equation. Students will realize that to create all these lines, they must work with the slope and y-intercept of the lines. 



Assigning practice prematurely can cause frustration, so, I'll assign Homework Systems and Parallel Lines only after satisfactorily completing the lesson. I may also modify the assignment by asking students to omit the last two word problems, or simply ask a student to do only certain Problems, #5, 6, and 7, for example. Eventually, all students should be able to do all 12 problems.