Stimulating Understanding of Computational science through Collaboration, Exploration, Experiment, and Discovery for students with Hearing Impairments
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For Teachers!

Systems of Linear Equations Modeling

Lesson 2

Focus and Review: 
Starting the lesson with a review problem from lesson one and an introduction to this lesson is a good idea. Have the
students graph these three systems of equations using the systems of equations calculator. 

1) x + y = 2 2) x + y = 2 3) x + y = 2 

2x + 3y = 3 2x + 2y =4 x + y = 3 

It is a good idea to have the students describe the graphs of each system. How are they the same? How are they
different? 

Each system consists of straight lines. In system 1 we have two intersecting lines. In system 2, there is only one line. In
system 3, the lines are parallel. 

Statement of Objectives: 
Let the students know that this is what they will learn in this lesson: a system of two linear equations can create two
intersecting lines, two parallel lines, or one line. 

Teacher Input: 
In the first example, the calculator told us that the system was consistent and independent. That means that the two equations
have different slopes and different y-intercepts. 

x + y = 2; y = -x + 2; slope = -1 

2x + 3y = 3; y = -2/3 + 1; slope = -2/3 

Explain to the students that different slopes create intersecting lines. A system is considered consistent when it has at
least one solution. A system is considered independent when it has exactly one solution. 
This system would be both independent and consistent. 

In the second example, the calculator shows us that this system is consistent and dependent. This means that the two equations
have the same slope and the same y-intercept. 

x + y = 2; y = -x + 2 ; slope = -1, y-intercept = 2 

2x + 2y = 4; y = -2x + 4; slope = -1, y-intercept = 2 
 

Because these two equations have the same slope as well as the same y-intercept, they are the same line. This system is a consistent system because it has at least one solution and it is a dependent system because it has an infinite number of solutions. 

In the third example, the calculator shows us that this system is inconsistent. We see that the two equations never meet. This is because the equations have the same slope and different y-intercepts. 

x + y = 2; y = -x + 2 slope = -1, y-intercept = 2 

x + y = 3; y = -x + 3 slope = -1, y-intercept = 3 

When lines have the same slope and different y-intercepts, they are parallel lines. Parallel lines never intersect. Because these lines never intersect, they have no solution. Since they have no solution, they are not consistent, dependent or independent like the last two systems we looked at. This system is inconsistent. A system with no solution is an inconsistent system. 

Guided Practice: 
Now let us try a few other examples: 

Determine whether  these systems are consistent and independent, consistent and dependent, or inconsistent. Use the systems of
equations calculator to verify your answers. 

1) 4x + 4 = 4y 2) 4x + 4 = 4y 3) 4x + 4 = 4y 
8x – 8y = 16 8x + 8y = 16 8x – 8y = -8 


How do you know that your answer is correct? The first thing that we need to do is to find the slope and y-intercept of
these equations. The best way to do this is to put these equations in slope- intercept form (solve for y). 

4x + 4 = 4y; y = x + 1; slope = 1, y-intercept =1 

8x –8y = 16; y = x –2; slope = 1, y-intercept = -2 

These two equations have the same slope, but different y-intercepts. That means that they are moving in the same
direction and doing the same thing, but at different locations. These lines are parallel. This system is an inconsistent
system because these lines are parallel and have no solution. 

2) 4x + 4 = 4y 
8x + 8y = 16 

Again, we need to find the slope and y-intercept of these equations. 

4x + 4 = 4y; y = x + 1; slope = 1, y-intercept =1 

8x + 8y = 16; y = -x +2; slope = -1, y-intercept = 2 

These two equations have different slopes and different y-intercepts. These are intersecting lines. This system is a consistent system because it has at least one solution and it is an independent system because it has exactly one solution. (It is a good idea to graph the system using colored pencils or chalk.) 

3) 4x + 4 = 4y 
8x – 8y = -8 

4x + 4 = 4y; y = x + 1; slope = 1, y-intercept = 1 

8x – 8y = 8; y = x + 1; slope = 1, y-intercept = 1 

These two equations have the same slope and the same y-intercept. These equations create one line. This system is an
inconsistent system because it has an infinite number of solutions. (It is a good idea to graph the system using colored
pencils or chalk.) 

Independent Practice: 
Determine if these systems are consistent and independent, consistent and dependent, or inconsistent. Use the systems of
equations calculator to find your answers. 

1) 8x – 12y = 10 2) 9x + 18 = 21y 3) 6x + 8y = 14 
6x – 9y = 15 2x + 4y = 22 3x + 4y = 7 

Answers to 1, 2, and 3. 

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