7.4: Special Types of Linear Systems
Learning Objectives
- Identify and understand what is meant by an inconsistent linear system.
- Identify and understand what is meant by a consistent linear system.
- Identify and understand what is meant by a dependent linear system.
Introduction
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 1
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 2
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 3
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.
Determining the Type of System Algebraically
A third option for identifying systems as consistent, inconsistent or dependent is to just solve the system and use the result as a guide.
Example 4
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
\begin{align*}10x - 3y &= 3\\ 2x + y &= 9\end{align*}
Solution
Let’s solve this system using the substitution method.
Solve the second equation for \begin{align*}y\end{align*}:
\begin{align*}2x + y = 9 \Rightarrow y = -2x + 9\end{align*}
Substitute that expression for \begin{align*}y\end{align*} in the first equation:
\begin{align*}10x - 3y &= 3 \\ 10x - 3(-2x + 9) &= 3\\ 10x + 6x - 27 &= 3\\ 16x &= 30\\ x &= \frac{15}{8}\end{align*}
Substitute the value of \begin{align*}x\end{align*} back into the second equation and solve for \begin{align*}y\end{align*}:
\begin{align*}2x + y = 9 \Rightarrow y = -2x + 9 \Rightarrow y = -2 \cdot \frac{15}{8} + 9 \Rightarrow y = \frac{21}{4}\end{align*} The solution to the system is \begin{align*}\left ( \frac{15}{8}, \frac{21}{4} \right )\end{align*}. The system is consistent since it has only one solution.
Example 5
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
\begin{align*}3x - 2y & = 4\\ 9x - 6y & = 1\end{align*}
Solution
Let’s solve this system by the method of multiplication.
Multiply the first equation by 3:
\begin{align*}3(3x - 2y = 4) \qquad \qquad \qquad \qquad 9x - 6y = 12\!\\ {\;} \qquad \qquad \qquad \qquad \Rightarrow \!\\ 9x - 6y = 1 \qquad \qquad \qquad \qquad \quad \ 9x - 6y = 1\end{align*}
Add the two equations:
\begin{align*}& \qquad 9x - 6y = 4\\ & \qquad \underline{9x - 6y = 1}\\ & \qquad \qquad \ \ 0 = 13 \quad \text{This statement is not true.}\end{align*}
If our solution to a system turns out to be a statement that is not true, then the system doesn’t really have a solution; it is inconsistent.
Example 6
Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.
\begin{align*}4x + y &= 3\\ 12x + 3y &= 9\end{align*}
Solution
Let’s solve this system by substitution.
Solve the first equation for \begin{align*}y\end{align*}:
\begin{align*}4x + y = 3 \Rightarrow y = -4x + 3\end{align*}
Substitute this expression for \begin{align*}y\end{align*} in the second equation:
\begin{align*}12x + 3y &= 9\\ 12x + 3(-4x + 3) &= 9\\ 12x - 12x + 9 &= 9\\ 9 &= 9\end{align*}
This statement is always true.
If our solution to a system turns out to be a statement that is always true, then the system is dependent.
A second glance at the system in this example reveals that the second equation is three times the first equation, so the two lines are identical. The system has an infinite number of solutions because they are really the same equation and trace out the same line.
Let’s clarify this statement. An infinite number of solutions does not mean that any ordered pair \begin{align*}(x, y)\end{align*} satisfies the system of equations. Only ordered pairs that solve the equation in the system (either one of the equations) are also solutions to the system. There are infinitely many of these solutions to the system because there are infinitely many points on any one line.
For example, (1, -1) is a solution to the system in this example, and so is (-1, 7). Each of them fits both the equations because both equations are really the same equation. But (3, 5) doesn’t fit either equation and is not a solution to the system.
In fact, for every \begin{align*}x-\end{align*}value there is just one \begin{align*}y-\end{align*}value that fits both equations, and for every \begin{align*}y-\end{align*}value there is exactly one \begin{align*}x-\end{align*}value—just as there is for a single line.
Let’s summarize how to determine the type of system we are dealing with algebraically.
- A consistent system will always give exactly one solution.
- An inconsistent system will yield a statement that is always false (like \begin{align*}0 = 13\end{align*}).
- A dependent system will yield a statement that is always true (like \begin{align*}9 = 9\end{align*}).
Applications
In this section, we’ll see how consistent, inconsistent and dependent systems might arise in real life.
Example 7
The movie rental store CineStar offers customers two choices. Customers can pay a yearly membership of $45 and then rent each movie for $2 or they can choose not to pay the membership fee and rent each movie for $3.50. How many movies would you have to rent before the membership becomes the cheaper option?
Solution
Let’s translate this problem into algebra. Since there are two different options to consider, we can write two different equations and form a system.
The choices are “membership” and “no membership.” We’ll call the number of movies you rent \begin{align*}x\end{align*} and the total cost of renting movies for a year \begin{align*}y\end{align*}.
flat fee | rental fee | total | |
---|---|---|---|
membership | $45 | \begin{align*}2x\end{align*} | \begin{align*}y = 45 + 2x\end{align*} |
no membership | $0 | \begin{align*}3.50x\end{align*} | \begin{align*}y = 3.5x\end{align*} |
The flat fee is the dollar amount you pay per year and the rental fee is the dollar amount you pay when you rent a movie. For the membership option the rental fee is \begin{align*}2x\end{align*}, since you would pay $2 for each movie you rented; for the no membership option the rental fee is \begin{align*}3.50x\end{align*}, since you would pay $3.50 for each movie you rented.
Our system of equations is:
\begin{align*}y = 45 + 2x\!\\ y = 3.50x\end{align*}
Here’s a graph of the system:
Now we need to find the exact intersection point. Since each equation is already solved for \begin{align*}y\end{align*}, we can easily solve the system with substitution. Substitute the second equation into the first one:
\begin{align*}y = 45 + 2x\!\\ {\;} \qquad \qquad \qquad \ \Rightarrow 3.50x = 45 + 2x \Rightarrow 1.50x = 45 \Rightarrow x = 30 \ \text{movies}\!\\ y = 3.50x\end{align*}
You would have to rent 30 movies per year before the membership becomes the better option.
This example shows a real situation where a consistent system of equations is useful in finding a solution. Remember that for a consistent system, the lines that make up the system intersect at single point. In other words, the lines are not parallel or the slopes are different.
In this case, the slopes of the lines represent the price of a rental per movie. The lines cross because the price of rental per movie is different for the two options in the problem
Now let’s look at a situation where the system is inconsistent. From the previous explanation, we can conclude that the lines will not intersect if the slopes are the same (and the \begin{align*}y-\end{align*}intercept is different). Let’s change the previous problem so that this is the case.
Example 8
Two movie rental stores are in competition. Movie House charges an annual membership of $30 and charges $3 per movie rental. Flicks for Cheap charges an annual membership of $15 and charges $3 per movie rental. After how many movie rentals would Movie House become the better option?
Solution
It should already be clear to see that Movie House will never become the better option, since its membership is more expensive and it charges the same amount per movie as Flicks for Cheap.
The lines on a graph that describe each option have different \begin{align*}y-\end{align*}intercepts—namely 30 for Movie House and 15 for Flicks for Cheap—but the same slope: 3 dollars per movie. This means that the lines are parallel and so the system is inconsistent.
Now let’s see how this works algebraically. Once again, we’ll call the number of movies you rent \begin{align*}x\end{align*} and the total cost of renting movies for a year \begin{align*}y\end{align*}.
flat fee | rental fee | total | |
---|---|---|---|
Movie House | $30 | \begin{align*}3x\end{align*} | \begin{align*}y = 30 + 3x\end{align*} |
Flicks for Cheap | $15 | \begin{align*}3x\end{align*} | \begin{align*}y = 15 + 3x\end{align*} |
The system of equations that describes this problem is:
\begin{align*}y = 30 + 3x\!\\ y = 15 + 3x\end{align*}
Let’s solve this system by substituting the second equation into the first equation:
\begin{align*}y = 30 + 3x\!\\ {\;} \qquad \qquad \qquad \quad \Rightarrow 15 + 3x = 30 + 3x \Rightarrow 15 = 30 \qquad \text{This statement is always false.}\!\\ y = 15 + 3x\end{align*}
This means that the system is inconsistent.
Example 9
Peter buys two apples and three bananas for $4. Nadia buys four apples and six bananas for $8 from the same store. How much does one banana and one apple costs?
Solution
We must write two equations: one for Peter’s purchase and one for Nadia’s purchase.
Let’s say \begin{align*}a\end{align*} is the cost of one apple and \begin{align*}b\end{align*} is the cost of one banana.
cost of apples | cost of bananas | total cost | |
---|---|---|---|
Peter | \begin{align*}2a\end{align*} | \begin{align*}3b\end{align*} | \begin{align*}2a + 3b = 4\end{align*} |
Nadia | \begin{align*}4a\end{align*} | \begin{align*}6b\end{align*} | \begin{align*}4a + 6b = 8\end{align*} |
The system of equations that describes this problem is:
\begin{align*}2a + 3b = 4\!\\ 4a + 6b = 8\end{align*}
Let’s solve this system by multiplying the first equation by -2 and adding the two equations:
\begin{align*}-2(2a + 3b = 4) \qquad \qquad \quad -4a - 6b = -8\!\\ {\;} \qquad \qquad \qquad \qquad \ \Rightarrow\!\\ \ 4a + 6b = 8 \qquad \qquad \qquad \qquad \underline{\;\;4a + 6b = 8\;\;}\!\\ {\;}\qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \ 0 + 0 = 0\end{align*}
This statement is always true. This means that the system is dependent.
Looking at the problem again, we can see that we were given exactly the same information in both statements. If Peter buys two apples and three bananas for $4, it makes sense that if Nadia buys twice as many apples (four apples) and twice as many bananas (six bananas) she will pay twice the price ($8). Since the second equation doesn’t give us any new information, it doesn’t make it possible to find out the price of each fruit.
Review Questions
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*}
Find the solution of each system of equations using the method of your choice. State if the system is inconsistent or dependent.
- \begin{align*}3x + 2y = 4\!\\ - 2x + 2y = 24\end{align*}
- \begin{align*}5x - 2y = 3\!\\ 2x - 3y = 10\end{align*}
- \begin{align*}3x - 4y = 13\!\\ y = -3x - 7\end{align*}
- \begin{align*}5x - 4y = 1\!\\ -10x + 8y = -30\end{align*}
- \begin{align*}4x + 5y = 0\!\\ 3x = 6y + 4.5\end{align*}
- \begin{align*}-2y + 4x = 8\!\\ y - 2x = -4\end{align*}
- \begin{align*}x - \frac{1}{2}y = \frac{3}{2}\!\\ 3x + y = 6\end{align*}
- \begin{align*}0.05x + 0.25y = 6\!\\ x + y = 24\end{align*}
- \begin{align*}x + \frac{2}{3}y = 6\!\\ 3x + 2y = 2\end{align*}
- A movie theater charges $4.50 for children and $8.00 for adults.
- On a certain day, 1200 people enter the theater and $8375 is collected. How many children and how many adults attended?
- The next day, the manager announces that she wants to see them take in $10000 in tickets. If there are 240 seats in the house and only five movie showings planned that day, is it possible to meet that goal?
- At the same theater, a 16-ounce soda costs $3 and a 32-ounce soda costs $5. If the theater sells 12,480 ounces of soda for $2100, how many people bought soda? (Note: Be careful in setting up this problem!)
- Jamal placed two orders with an internet clothing store. The first order was for 13 ties and 4 pairs of suspenders, and totaled $487. The second order was for 6 ties and 2 pairs of suspenders, and totaled $232. The bill does not list the per-item price, but all ties have the same price and all suspenders have the same price. What is the cost of one tie and of one pair of suspenders?
- An airplane took four hours to fly 2400 miles in the direction of the jet-stream. The return trip against the jet-stream took five hours. What were the airplane’s speed in still air and the jet-stream's speed?
- Nadia told Peter that she went to the farmer’s market and bought two apples and one banana, and that it cost her $2.50. She thought that Peter might like some fruit, so she went back to the seller and bought four more apples and two more bananas. Peter thanked Nadia, but told her that he did not like bananas, so he would only pay her for four apples. Nadia told him that the second time she paid $6.00 for the fruit.
- What did Peter find when he tried to figure out the price of four apples?
- Nadia then told Peter she had made a mistake, and she actually paid $5.00 on her second trip. Now what answer did Peter get when he tried to figure out how much to pay her?
- Alicia then showed up and told them she had just bought 3 apples and 2 bananas from the same seller for $4.25. Now how much should Peter pay Nadia for four apples?