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Determining the Number of Solutions of a Linear System

Determining 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 the examples in the previous lesson except the last one all had two intersecting lines. Each system had one solution.

A system with parallel lines, like the last example from the previous lesson, has no solution. What happened in that last example? The equations have coincident lines, and so the system had infinitely many solutions.

We’ll organize these results in the figure 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 below which we solved in the previous lesson.

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\begin{array}{ccc}\hfill y& =\hfill & \frac{1}{2}x-3\hfill \\ \hfill x-2y& =\hfill & 4\hfill \end{array}\hfill \\ \text{The first line is in slope–intercept form.}\hfill & & & \text{If we solve the second equation for}\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}\text{we get}\hfill \\ \hfill y=\frac{1}{2}x-3\hfill & & & \hfill \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\hfill x-2y& =\hfill & 4\hfill \\ \hfill -2y& =\hfill & \text{−}x+4\hfill \\ \hfill y& =\hfill & \frac{1}{2}x-2\hfill \end{array}\hfill \\ \hfill m=\frac{1}{2},b=-3\hfill & & & \hfill m=\frac{1}{2},b=-2\hfill \end{array}\)

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

The figure below 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 the example from the previous lesson that gave us parallel lines.

\(\begin{array}{c}y=\frac{1}{2}x-3\hfill \\ x-2y=4\hfill \end{array}\)

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

\(y=\frac{1}{2}x-3\phantom{\rule{2em}{0ex}}y=\frac{1}{2}x-2\)

Do you recognize that it is impossible to have a single ordered pair \(\left(x,y\right)\) 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 the two figures below.

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.

Example

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{array}{c}y=3x-1\hfill \\ 6x-2y=12\hfill \end{array}.\)

Solution

\(\begin{array}{cccc}\begin{array}{c}\text{We will compare the slopes and intercepts}\hfill \\ \text{of the two lines.}\hfill \\ \text{The first equation is already in}\hfill \\ \text{slope-intercept form.}\hfill \\ \\ \text{Write the second equation in}\hfill \\ \text{slope–intercept form.}\hfill \end{array}\hfill & & & \begin{array}{}\\ \\ \begin{array}{ccc}\hfill y& =\hfill & 3x-1\hfill \\ \hfill 6x-2y& =\hfill & 12\hfill \end{array}\hfill \\ \\ \begin{array}{ccc}\phantom{\rule{3em}{0ex}}y\hfill & =\hfill & 3x-1\hfill \end{array}\hfill \\ \phantom{\rule{0.3em}{0ex}}\begin{array}{}\\ \\ \hfill 6x-2y& =\hfill & 12\hfill \\ \hfill -2y& =\hfill & -6x+12\hfill \\ \hfill \frac{-2y}{-2}& =\hfill & \frac{-6x+12}{-2}\hfill \end{array}\hfill \end{array}\hfill \\ \text{Find the slope and intercept of each line.}\hfill & & & \hfill \begin{array}{ccccccccc}\hfill y& =\hfill & 3x-6\hfill & & & & & & \\ \\ \hfill y& =\hfill & 3x-1\hfill & & & & \hfill y& =\hfill & 3x-6\hfill \\ \hfill m& =\hfill & 3\hfill & & & & \hfill m& =\hfill & 3\hfill \\ \hfill b& =\hfill & -1\hfill & & & & \hfill b& =\hfill & -6\hfill \end{array}\hfill \\ & & & \begin{array}{c}\text{Since the slopes are the same and}\phantom{\rule{0.2em}{0ex}}y\text{-intercepts}\hfill \\ \text{are different, the lines are parallel.}\hfill \end{array}\hfill \end{array}\)

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

Example

Without graphing, determine the number of solutions and then classify the system of equations: \(\begin{array}{c}2x+y=-3\hfill \\ x-5y=5\hfill \end{array}.\)

Solution

\(\begin{array}{cccc}\begin{array}{c}\text{We will compare the slope and intercepts of the two lines.}\hfill \\ \\ \end{array}\hfill & & & \phantom{\rule{6em}{0ex}}\begin{array}{ccc}\hfill 2x+y& =\hfill & -3\hfill \\ \hfill x-5y& =\hfill & 5\hfill \end{array}\hfill \\ \\ \begin{array}{c}\text{Write both equations in slope–intercept form.}\hfill \\ \\ \end{array}\hfill & & & \hfill \begin{array}{cccccccc}\hfill 2x+y& =\hfill & -3\hfill & & & \hfill x-5y& =\hfill & 5\hfill \\ \hfill y& =\hfill & -2x-3\hfill & & & \hfill -5y& =\hfill & \text{−}x+5\hfill \\ & & & & & \hfill \frac{-5y}{-5}& =\hfill & \frac{\text{−}x+5}{-5}\hfill \\ & & & & & \hfill y& =\hfill & \frac{1}{5}x-1\hfill \end{array}\hfill \\ \\ \begin{array}{c}\text{Find the slope and intercept of each line.}\hfill \\ \\ \end{array}\hfill & & & \hfill \phantom{\rule{0.5em}{0ex}}\begin{array}{cccccccccc}\hfill y& =\hfill & -2x-3\hfill & & & & & \hfill \phantom{\rule{2em}{0ex}}y& =\hfill & \frac{1}{5}x-1\hfill \\ \hfill m& =\hfill & -2\hfill & & & & & \hfill \phantom{\rule{2em}{0ex}}m& =\hfill & \frac{1}{5}\hfill \\ \hfill b& =\hfill & -3\hfill & & & & & \hfill \phantom{\rule{2em}{0ex}}b& =\hfill & -1\hfill \end{array}\hfill \\ & & & \text{Since the slopes are different, the lines intersect.}\hfill \end{array}\)

A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

Example

Without graphing, determine the number of solutions and then classify the system of equations. \(\begin{array}{c}3x-2y=4\hfill \\ y=\frac{3}{2}x-2\hfill \end{array}\)

Solution

\(\begin{array}{cccc}\begin{array}{c}\text{We will compare the slope and intercepts of the two lines.}\hfill \\ \\ \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill 3x-2y& =\hfill & 4\hfill \\ \hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ \\ \begin{array}{c}\text{Write the first equation in slope–intercept form.}\hfill \\ \\ \end{array}\hfill & & & \hfill \phantom{\rule{1.3em}{0ex}}\begin{array}{ccc}\hfill 3x-2y& =\hfill & 4\hfill \\ \hfill -2y& =\hfill & -3x+4\hfill \\ \hfill \frac{-2y}{-2}& =\hfill & \frac{-3x+4}{-2}\hfill \\ \hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ \\ \begin{array}{c}\text{The second equation is already in}\hfill \\ \text{slope–intercept form.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{3em}{0ex}}\begin{array}{ccc}\hfill y& =\hfill & \frac{3}{2}x-2\hfill \end{array}\hfill \\ & & & \begin{array}{c}\text{Since the equations are the same, they have the same slope}\hfill \\ \text{and same}\phantom{\rule{0.2em}{0ex}}y\text{-intercept and so the lines are coincident.}\hfill \end{array}\hfill \end{array}\)

A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

This lesson is part of:

Systems of Linear Equations I

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