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Determining the Type of Linear System

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Solving Systems with No or Infinitely Many Solutions Using Graphing
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At a craft fair, Jamal bought two wooden crafts and one needlework craft for a total of $21. At the same craft fair, Kenia bought two needwork crafts and four wooden crafts for a total of $28. Did Jamal and Kenia buy the same items? Why or why not? Assume no discounts were offered.

Guidance

So far we have looked at linear systems of equations in which the lines always intersected in one, unique point. What happens if this is not the case? What could the graph of the two lines look like? In Examples A and B below we will explore the two possibilities.

Example A

Graph the system:

y &= 2x-5\\y &= 2x+4

Solution:

In this example both lines have the same slope but different y- intercepts. When graphed, they are parallel lines and never intersect. This system has no solution. Another way to say this is to say that it is inconsistent .

Example B

Graph the system:

2x-3y &= 6\\-4x + 6y &= -12

Solution:

In this example both lines have the same slope and y- intercept. This is more apparent when the equations are written in slope intercept form:

 y = \frac{2}{3}x-2 \ and \ y = \frac{2}{3}x-2

When we graph them, they are one line, coincident , meaning they have all points in common. This means that there are an infinite number of solutions to the system. Because this system has at least one solution it is considered to be consistent .

Consistent systems are systems which have at least one solution. If the system has exactly one, unique solution then it is independent . All of the systems we solved in the last section were independent. If the system has infinite solutions, like the system in Example B, then it is called dependent .

Example C

Classify the following system:

10x-2y &= 10\\y &= 5x-5

Solution:

Rearranging the first equation into slope intercept form we get y=5x-5 , which is exactly the same as the second equation. This means that they are the same line. Therefore the system is consistent and dependent.

Intro Problem Revisit The system of linear equations represented by this situation is: 2w + n &= 21\\4w + 2n &= 28

If you graph these two linear equations on the same graph, there is no point of intersection. Therefore there is no solution and Jamal and Kenia couldn't have bought the same items.

Guided Practice

Classify the following systems as consistent, inconsistent, independent or dependent. You may do this with or without graphing them. You do not need to find the unique solution if there is one.

1. 5x-y &= 15\\x+5y &= 15

2. 9x-12y &= -24\\-3x+4y &= 8

3. 6x+8y &= 12\\-3x-4y &= 10

Answers

1. The first step is to rearrange both equations into slope intercept form so that we can compare these attributes.

5x-y &= 15 \rightarrow y=5x-15\\x+5y &= 15 \rightarrow y= - \frac{1}{5}x+3

The slopes are not the same so the lines are neither parallel nor coincident. Therefore, the lines must intersect in one point. The system is consistent and independent.

2. Again, rearrange the equations into slope intercept form:

9x-12y = -24 & \rightarrow y=\frac{3}{4}x+2 \\-3x+4y = 8 & \rightarrow y=\frac{3}{4}x+2

Now, we can see that both the slope and the y- intercept are the same and therefore the lines are coincident. The system is consistent and dependent.

3. The equations can be rewritten as follows:

6x+8y &= 12 \rightarrow y= - \frac{3}{4}x+\frac{3}{2} \\-3x-4y &= 10 \rightarrow y= - \frac{3}{4}x-\frac{5}{2}

In this system the lines have the same slope but different y- intercepts so they are parallel lines. Therefore the system is inconsistent. There is no solution.

Vocabulary

Parallel
Two or more lines in the same plane that never intersect. They have the same slope and different y- intercepts.
Coincident
Lines which have all points in common. They are line which “coincide” with one another or are the same line.
Consistent
Describes a system with at least one solution.
Inconsistent
Describes a system with no solution.
Dependent
Describes a consistent system with infinite solutions.
Independent
Describes a consistent system with exactly one solution.

Practice

Describe the systems graphed below both algebraically (consistent, inconsistent, dependent, independent) and geometrically (intersecting lines, parallel lines, coincident lines).

Classify the following systems as consistent, inconsistent, independent or dependent. You may do this with or without graphing them. You do not need to find the unique solution if there is one.

  1. .
4x-y &= 8\\y &= 4x+3
  1. .
5x+y &= 10\\y &= 5x+10
  1. .
2x-2y &= 11\\y &= x+13
  1. .
-7x+3y &= -21\\14x-6y &= 42
  1. .
y &= - \frac{3}{5}x+1\\3x+5y &= 5
  1. .
6x-y &= 18\\y &= \frac{1}{6}x+3

In problems 10-15 you will be writing your own systems. Your equations should be in standard form, Ax+By=C . Try to make them look different even if they are the same equation.

  1. Write a system which is consistent and independent.
  2. Write a system which is consistent and dependent.
  3. Write a system which is inconsistent.
  4. Write a system where the solution is (-1, 2), one line is vertical and the second is horizontal.
  5. Write a system where the solution is (-1, 2), one line is vertical or horizontal and the second is neither.
  6. Write a system where the solution is (-1, 2) and neither line is vertical nor horizontal.

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