7.7: Consistent and Inconsistent Linear Systems
What if you were given a system of equations like \begin{align*}2x - y = 5\end{align*} and \begin{align*}10x - 5y = 25\end{align*}? How could you rewrite these equations to determine the number of solutions the system has? After completing this Concept, you'll be able to identify whether a system of equations like this one is an inconsistent one, a consistent one, or a dependent one.
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CK-12 Foundation: 0707S Special Types of Linear Systems (H264)
Guidance
As we saw in Section 7.1, a system of linear equations is a set of linear equations which must be solved together. The lines in the system can be graphed together on the same coordinate graph and the solution to the system is the point at which the two lines intersect.
Or at least that’s what usually happens. But what if the lines turn out to be parallel when we graph them?
If the lines are parallel, they won’t ever intersect. That means that the system of equations they represent has no solution. A system with no solutions is called an inconsistent system.
And what if the lines turn out to be identical?
If the two lines are the same, then every point on one line is also on the other line, so every point on the line is a solution to the system. The system has an infinite number of solutions, and the two equations are really just different forms of the same equation. Such a system is called a dependent system.
But usually, two lines cross at exactly one point and the system has exactly one solution:
A system with exactly one solution is called a consistent system.
To identify a system as consistent, inconsistent, or dependent, we can graph the two lines on the same graph and see if they intersect, are parallel, or are the same line. But sometimes it is hard to tell whether two lines are parallel just by looking at a roughly sketched graph.
Another option is to write each line in slope-intercept form and compare the slopes and \begin{align*}y-\end{align*} intercepts of the two lines. To do this we must remember that:
- Lines with different slopes always intersect.
- Lines with the same slope but different \begin{align*}y-\end{align*}intercepts are parallel.
- Lines with the same slope and the same \begin{align*}y-\end{align*}intercepts are identical.
Example A
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
\begin{align*}2x - 5y &= 2\\ 4x + y &= 5\end{align*}
Solution
We must rewrite the equations so they are in slope-intercept form
\begin{align*}2x - 5y = 2 \qquad \qquad \qquad \qquad -5y = -2x + 2 \qquad \qquad \qquad \qquad y = \frac{2}{5}x - \frac{2}{5}\!\\ {\;} \qquad \qquad \qquad \qquad \Rightarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow \!\\\ 4x + y = 5 \qquad \qquad \qquad \qquad \quad \ y = -4x + 5 \qquad \qquad \qquad \qquad \quad y = -4x + 5\end{align*}
The slopes of the two equations are different; therefore the lines must cross at a single point and the system has exactly one solution. This is a consistent system.
Example B
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
\begin{align*}3x &= 5 - 4y\\ 6x + 8y &= 7\end{align*}
Solution
We must rewrite the equations so they are in slope-intercept form
\begin{align*}3x = 5 - 4y \qquad \qquad \qquad \qquad \quad 4y = -3x + 5 \qquad \qquad \qquad \qquad y = - \frac{3}{4}x + \frac{5}{4}\!\\ {\;} \qquad \qquad \qquad \qquad \Rightarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow \!\\ 6x + 8y = 7 \qquad \qquad \qquad \qquad \quad 8y = -6x + 7 \qquad \qquad \qquad \qquad y = - \frac{3}{4}x + \frac{7}{8}\end{align*}
The slopes of the two equations are the same but the \begin{align*}y-\end{align*}intercepts are different; therefore the lines are parallel and the system has no solutions. This is an inconsistent system.
Example C
Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.
\begin{align*}x + y &= 3\\ 3x + 3y &= 9\end{align*} Solution
We must rewrite the equations so they are in slope-intercept form
\begin{align*}x + y = 3 \qquad \qquad \qquad \qquad \qquad y = -x + 3 \qquad \qquad \qquad \qquad \qquad y = -x + 3\!\\ {\;} \qquad \qquad \qquad \qquad \Rightarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow \!\\\ 3x + 3y = 9 \qquad \qquad \qquad \qquad \quad 3y = -3x + 9 \qquad \qquad \qquad \qquad \quad y = -x + 3\end{align*}
The lines are identical; therefore the system has an infinite number of solutions. It is a dependent system.
Watch this video for help with the Examples above.
CK-12 Foundation: Special Types of Linear Systems
Vocabulary
- A system with no solutions is called an inconsistent system. For linear equations, this occurs with parallel lines.
- A system where the two equations overlap at one, multiple, or infinitely many points is called a consistent system.
- Coincident lines are lines with the same slope and \begin{align*}y-\end{align*}intercept. The lines completely overlap.
- When solving a system of coincident lines, the resulting equation will be without variables and the statement will be true. You can conclude the system has an infinite number of solutions. This is called a consistent-dependent system.
Guided Practice
Determine whether the following system of linear equations has zero, one, or infinitely many solutions:
\begin{align*}\begin{cases} 2y+6x=20\\ y=-3x+7 \end{cases}\end{align*}
What kind of system is this?
Solution:
It is easier to compare equations when they are in the same form. We will rewrite the first equation in slope-intercept form.
\begin{align*}2y+6x=20 \Rightarrow y+3x=10 \Rightarrow y=-3x+10\end{align*}
Since the two equations have the same slope, but different \begin{align*}y\end{align*}-intercepts, they are different but parallel lines. Parallel lines never intersect, so they have no solutions.
Since the lines are parallel, it is an inconsistent system.
Practice
Express each equation in slope-intercept form. Without graphing, state whether the system of equations is consistent, inconsistent or dependent.
- \begin{align*}3x - 4y = 13\!\\ y = -3x - 7\end{align*}
- \begin{align*}\frac{3}{5}x + y = 3\!\\ 1.2x + 2y = 6\end{align*}
- \begin{align*}3x - 4y = 13\!\\ y = -3x - 7\end{align*}
- \begin{align*}3x - 3y = 3\!\\ x - y = 1\end{align*}
- \begin{align*}0.5x - y = 30\!\\ 0.5x - y = -30\end{align*}
- \begin{align*}4x - 2y = -2\!\\ 3x + 2y = -12\end{align*}
- \begin{align*}3x + y = 4\!\\ y = 5 - 3x\end{align*}
- \begin{align*}x - 2y = 7\!\\ 4y - 2x = 14\end{align*}
- \begin{align*}&-2y+4x=8\\ &y-2x=-4\end{align*}
- \begin{align*}&x-\frac{y}{2}=\frac{3}{2}\\ &3x+y=6\end{align*}
- \begin{align*}&0.05x+0.25y=6\\ &x+y=24\end{align*}
- \begin{align*}&x+\frac{2y}{3}=6\\ &3x+2y=2\end{align*}
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Here you'll learn the difference between three special types of linear systems: inconsistent linear systems, consistent linear systems, and dependent linear systems. You'll then use that information to determine the number of solutions a system has.