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5.1 Solve systems of equations by graphing  (Page 4/14)

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Coincident lines

Coincident lines have the same slope and same y -intercept.

 

Determine the number of solutions of a linear system

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in [link] through [link] all had two intersecting lines. Each system had one solution.

A system with parallel lines, like [link] , has no solution. What happened in [link] ? The equations have coincident lines    , and so the system had infinitely many solutions.

We’ll organize these results in [link] below:

This table has two columns and four rows. The first row labels each column “Graph” and “Number of solutions.” Under “Graph” are “2 intersecting lines,” “Parallel lines,” and “Same line.” Under “Number of solutions” are “1,” “None,” and “Infinitely many.”

Parallel lines have the same slope but different y -intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in [link] .

{ y = 1 2 x − 3 x − 2 y = 4 The first line is in slope–intercept form. If we solve the second equation for y , we get y = 1 2 x − 3 x − 2 y = 4 −2 y = − x + 4 y = 1 2 x − 2 m = 1 2 , b = −3 m = 1 2 , b = −2

The two lines have the same slope but different y -intercepts. They are parallel lines.

[link] shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

This table is entitled “Number of Solutions of a Linear System of Equations.” There are four columns. The columns are labeled, “Slopes,” “Intercepts,” “Type of Lines,” “Number of Solutions.” Under “Slopes” are “Different,” “Same,” and “Same.” Under “Intercepts,” the first cell is blank, then the words “Different” and “Same” appear. Under “Types of Lines” are the words, “Intersecting,” “Parallel,” and “Coincident.” Under “Number of Solutions” are “1 point,” “No Solution,” and “Infinitely many solutions.”

Let’s take one more look at our equations in [link] that gave us parallel lines.

{ y = 1 2 x − 3 x − 2 y = 4

When both lines were in slope-intercept form we had:

y = 1 2 x − 3 y = 1 2 x − 2

Do you recognize that it is impossible to have a single ordered pair ( x , y ) that is a solution to both of those equations?

We call a system of equations like this an inconsistent system . It has no solution.

A system of equations that has at least one solution is called a consistent system    .

Consistent and inconsistent systems

A consistent system    of equations is a system of equations with at least one solution.

An inconsistent system    of equations is a system of equations with no solution.

We also categorize the equations in a system of equations by calling the equations independent or dependent . If two equations are independent equations    , they each have their own set of solutions. Intersecting lines and parallel lines are independent.

If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations    , we get coincident lines.

Independent and dependent equations

Two equations are independent if they have different solutions.

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

Let’s sum this up by looking at the graphs of the three types of systems. See [link] and [link] .

This figure shows three x y coordinate planes in a horizontal row. The first shows two lines intersecting. The second shows two parallel lines. The third shows two coincident lines.
This table has four columns and four rows. The columns are labeled, “Lines,” “Intersecting,” “Parallel,” and “Coincident.” In the first row under the labeled column “lines” it reads “Number of solutions.” Reading across, it tell us that an intersecting line contains 1 point, a parallel line provides no solution, and a coincident line has infinitely many solutions. A consistent/inconsistent line has consistent lines if they are intersecting, inconsistent lines if they are parallel and consistent if the lines are coincident. Finally, dependent and independent lines are considered independent if the lines intersect, they are also independent if the lines are parallel, and they are dependent if the lines are coincident.

Without graphing, determine the number of solutions and then classify the system of equations: { y = 3 x − 1 6 x − 2 y = 12 .

Solution

We will compare the slopes and intercepts of the two lines. The first equation is already in slope-intercept form. Write the second equation in slope–intercept form. { y = 3 x − 1 6 x − 2 y = 12 y = 3 x − 1 6 x − 2 y = 12 −2 y = −6 x + 12 −2 y −2 = −6 x + 12 −2 Find the slope and intercept of each line. y = 3 x − 6 y = 3 x − 1 y = 3 x − 6 m = 3 m = 3 b = −1 b = −6 Since the slopes are the same and y -intercepts are different, the lines are parallel.

A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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