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Determining the Type of Linear System

Identify consistent, inconsistent, and dependent systems

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Solving Systems with No or Infinitely Many Solutions Using Graphing

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.

Systems with No or Infinitely Many Solutions Using Graphing

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?

Let's graph the following systems.

  1.    

\begin{align*}y &= 2x-5\\ y &= 2x+4\end{align*}yy=2x5=2x+4

In this problem both lines have the same slope but different \begin{align*}y-\end{align*}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.

  1.     

\begin{align*}2x-3y &= 6\\ -4x + 6y &= -12\end{align*}

In this problem both lines have the same slope and \begin{align*}y-\end{align*}intercept. This is more apparent when the equations are written in slope intercept form:

\begin{align*} y = \frac{2}{3}x-2 \ and \ y = \frac{2}{3}x-2\end{align*}

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. If the system has infinite solutions, then it is called dependent.

Let's classify the following system:

\begin{align*}10x-2y &= 10\\ y &= 5x-5\end{align*}

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

Examples

Example 1

Earlier, you were asked if Jamal and Kenia bought the same items. 

The system of linear equations represented by this situation is: \begin{align*}2w + n &= 21\\ 4w + 2n &= 28\end{align*}

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.

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.

Example 2
\begin{align*}5x-y &= 15\\ x+5y &= 15\end{align*}

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

\begin{align*}5x-y &= 15 \rightarrow y=5x-15\\ x+5y &= 15 \rightarrow y= - \frac{1}{5}x+3\end{align*}

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.

Example 3
\begin{align*}9x-12y &= -24\\ -3x+4y &= 8\end{align*}

Again, rearrange the equations into slope intercept form:

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

Now, we can see that both the slope and the \begin{align*}y-\end{align*}intercept are the same and therefore the lines are coincident. The system is consistent and dependent.

Example 4
\begin{align*}6x+8y &= 12\\ -3x-4y &= 10\end{align*}

The equations can be rewritten as follows:

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

In this system the lines have the same slope but different \begin{align*}y-\end{align*}intercepts so they are parallel lines. Therefore the system is inconsistent. There is no solution.

Review

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

In problems 10-15 you will be writing your own systems. Your equations should be in standard form, \begin{align*}Ax+By=C\end{align*}. 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.

Answers for Review Problems

To see the Review Answers, open this PDF file and look for section 3.3. 

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Vocabulary

Coincident

Coincident lines have all points in common. They are lines which “coincide” with one another or are the same line.

Consistent

A system of equations is consistent if it has at least one solution.

Dependent

A system of equations is dependent if every solution for one equation is a solution for the other(s).

Inconsistent

A system of equations is inconsistent if it has no solutions.

Independent

A system of equations is independent if it has exactly one solution.

Parallel

Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.

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