Systems of Linear Equations Modeling

Let's start this lesson with a review problem from lesson one and an introduction to this lesson. Reviewing is always a good idea. Graph these three systems of equations, using the systems of equations calculator,or the slope form calculator., and record your findings.

Describe the graphs of each system. How are they the same? How are they different?systems of equations calculator,or the slope form calculator.

They are the same because each system consists of straight lines.
The differences are:
In system 1 we have two intersecting lines.
In systems 2 there is only one line.
In system 3, the lines are parallel.

In this lesson we will explore how a system of two linear equations can create two intersecting lines, two parallel lines, or one line.

Let's look at those three systems again. It is important to solve each equation for y and identify the slope and y-intercept.

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

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

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

Anytime lines have different slopes they create intersecting lines. A system with intersecting lines is considered consistent because it has at least one solution, and independent because it has exactly one solution. This system would be both independent and consistent.

In the second example, the calculator shows us this system is consistent and dependent. This means the two equations with 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 also a dependent system because it has an infinite number of solutions.

In the third example, the calculator shows us this system is inconsistent. We see 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 intercept. Because these lines never intercept, 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.

Now let us try a few other examples:

Tell if these systems are consistent and independent, consistent and dependent, or inconsistent. Use the systems of equations calculator,or the slope form calculator. to find your answers.

1) 4x + 4 = 4y 2) 4x + 4 = 4y 3) 4x + 4 = 4y

8x - 8y = 16 8x + 8y = 16 8x - 8y = -8

Let's look at the first system.

4x + 4 = 4y
8x - 8y = 16

How do you know if your answer is correct? The first thing 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). Then graph each system on a coordinate plane.

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

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

Let's look at the first system. 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, let's check if your answer is correct. We need to find the slope and y-intercept of these equations and graph the system on a coordinate plane.

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 independent system because it has exactly one solution.

3) 4x + 4 = 4y
8x - 8y = -8

Start by checking your answers. Find the slope and y-intercept of these equations and graph the system on a coordinate plane.

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.

Tell if these systems are consistent and independent, consistent and dependent, or inconsistent.

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

You may test your answers using the systems of equations calculator,or the slope form calculator.

You are ready to take a quiz.

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