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# 7.5: Special Types of Linear Systems

Difficulty Level: At Grade Created by: CK-12

Solutions to a system can have several forms:

• One intersection
• Two or more solutions
• No solutions
• An infinite amount of solutions

## Inconsistent Systems

This lesson will focus on the last two situations: systems with no solutions or systems with an infinite amount of solutions.

A system with parallel lines will have no solutions.

Remember from chapter 5 that parallel lines have the same slope. When graphed, the lines will have the same steepness with different \begin{align*}y-\end{align*}intercepts. Therefore, parallel lines will never intersect, thus they have no solution.

Algebraically, a system with no solutions looks like this when solved.

\begin{align*}\begin{cases} 4y=5-3x\\ 6x+8y=7 \end{cases}\end{align*}

The first equation in this system is “almost” solved for \begin{align*}y\end{align*}. Substitution would be appropriate to solve this system.

\begin{align*}\begin{cases} 4y=5-3x\\ 6x+8y=7 \end{cases} \rightarrow \quad \begin{cases} y=\frac{5}{4}-\frac{3}{4} x\\ 6x+8y=7 \end{cases}\end{align*}

Using the Substitution Property, replace the \begin{align*}y-\end{align*}variable in the second equation with its algebraic expression in equation #1.

\begin{align*}&&6x+8\left ( \frac{5}{4}-\frac{3}{4} x \right )&=7\\ \text{Apply the Distributive Property.} && 6x+10-6x&=7\\ \text{Add like terms.} && 10&=7\end{align*}

You have solved the equation correctly, yet the answer does not make sense.

When solving a system of parallel lines, the final equation will be untrue.

Because \begin{align*}10 \neq 7\end{align*} and you have done your math correctly, you can say this system has “no solutions.”

A system with no solutions is called an inconsistent system.

## Consistent Systems

Consistent systems, on the contrary, have at least one solution. This means there is at least one intersection of the lines. There are three cases for consistent systems:

• One intersection, as you have practiced the majority of this chapter
• Two or more intersections, as you will see when a quadratic equation intersects a linear equation
• Infinitely many intersections, as with coincident lines

Coincident lines are lines with the same slope and \begin{align*}y-\end{align*}intercept. The lines completely overlap.

When solving a consistent system involving coincident lines, the solution has the following result.

\begin{align*}\begin{cases} x+y=3\\ 3x+3y=9 \end{cases}\end{align*}

Multiply the first equation by –3: \begin{align*}\begin{cases} -3(x+y=3)\\ 3x+3y=9\end{cases} \rightarrow \quad \begin{cases} -3x-3y=-9\\ 3x+3y=9 \end{cases}\end{align*}

\begin{align*}0=0\end{align*}

There are no variables left and you KNOW you did the math correctly. However, this is a true statement.

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.

Example 1: Identify the system as consistent, inconsistent, or consistent-dependent.

\begin{align*}3x-2y&=4\\ 9x-6y&=1\end{align*}

Solution: Because both equations are in standard form, elimination is the best method to solve this system.

Multiply the first equation by 3.

\begin{align*}3(3x-2y=4)&&9x-6y=12\\ & \qquad \qquad \qquad \qquad \Rightarrow & \qquad\\ 9x-6y=1&&9x-6y=1\end{align*}

Subtract the two equations.

\begin{align*}& \ \ 9x-6y=12\\ & \underline{\;\; 9x-6y=1 \;\;}\\ & \qquad \quad \ 0=11 \quad \text{This Statement is not true.}\end{align*}

This is an untrue statement; therefore, you can conclude:

1. These lines are parallel.
2. The system has no solution.
3. The system is inconsistent.

Example 2: 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 move as Flicks for Cheap.

The lines that describe each option have different \begin{align*}y-\end{align*}intercepts, namely 30 for Movie House and 15 for Flicks for Cheap. They have the same slope, three dollars per movie. This means that the lines are parallel and the system is inconsistent.

Let’s see how this works algebraically.

Define the variables: Let \begin{align*}x=\end{align*} number of movies rented and \begin{align*}y=\end{align*} total rental cost

\begin{align*}\begin{cases} y=30+3x\\ y=15+3x \end{cases}\end{align*}

Because both equations are in slope-intercept form, solve this system by substituting the second equation into the first equation.

\begin{align*}15+3x=30+3x\Rightarrow15=30\end{align*}

This statement is always false. Therefore, the system is inconsistent with no solutions.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

1. Define an inconsistent system. What is true about these systems?
2. What are the three types of consistent systems?
3. You graph a system and see only one line. What can you conclude?
4. You graph a system and see the lines have an intersection point. What can you conclude?
5. The lines you graphed appear parallel. How can you verify the system will have no solution?
6. You graph a system and obtain the following graph. Is the system consistent or inconsistent? How many solutions does the system have?

In 7 – 24, find the solution of each system of equations using the method of your choice. Please state whether the system is inconsistent, consistent, or consistent-dependent.

1. \begin{align*}3x-4y=13\!\\ y=-3x-7\end{align*}
2. \begin{align*}4x+y=3\!\\ 12x+3y=9\end{align*}
3. \begin{align*}10x-3y=3\!\\ 2x+y=9\end{align*}
4. \begin{align*}2x-5y=2\!\\ 4x+y=5\end{align*}
5. \begin{align*}\frac{3x}{5}+y=3\!\\ 1.2x+2y=6\end{align*}
6. \begin{align*}3x-4y=13\!\\ y=-3x-7\end{align*}
7. \begin{align*}3x-3y=3\!\\ x-y=1\end{align*}
8. \begin{align*}0.5x-y=30\!\\ 0.5x-y=-30\end{align*}
9. \begin{align*}4x-2y=-2\!\\ 3x+2y=-12\end{align*}
10. \begin{align*}3x+2y=4\!\\ -2x+2y=24\end{align*}
11. \begin{align*}5x-2y=3\!\\ 2x-3y=10\end{align*}
12. \begin{align*}3x-4y=13\!\\ y=-3x-y\end{align*}
13. \begin{align*}5x-4y=1\!\\ -10x+8y=-30\end{align*}
14. \begin{align*}4x+5y=0\!\\ 3x=6y+4.5\end{align*}
15. \begin{align*}-2y+4x=8\!\\ y-2x=-4\end{align*}
16. \begin{align*}x-\frac{y}{2}=\frac{3}{2}\!\\ 3x+y=6\end{align*}
17. \begin{align*}0.05x+0.25y=6\!\\ x+y=24\end{align*}
18. \begin{align*}x+\frac{2y}{3}=6\!\\ 3x+2y=2\end{align*}
19. 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?
20. A 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 membership becomes the cheaper option? 21. A movie house charges$4.50 for children and $8.00 for adults. On a certain day, 1200 people enter the movie house and$8,375 is collected. How many children and how many adults attended?
22. Andrew placed two orders with an internet clothing store. The first order was for 13 ties and four pairs of suspenders, and it totaled $487. The second order was for six ties and two pairs of suspenders, and it 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?
23. 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?
24. Nadia told Peter that she went to the farmer’s market, that she 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 he told her that he did not like bananas, so he would pay her for only four apples. Nadia told him that the second time she paid$6.00 for the fruit. Please help Peter figure out how much to pay Nadia for four apples.

Mixed Review

1. A football stadium sells regular and box seating. There are twelve times as many regular seats as there are box seats. The total capacity of the stadium is 10,413. How many box seats are in the stadium? How many regular seats?
2. Find an equation for the line perpendicular to \begin{align*}y=-\frac{3}{5} x-8.5\end{align*} containing the point (2, 7).
3. Rewrite in standard form: \begin{align*}y=\frac{1}{6} x-4\end{align*}.
4. Find the sum: \begin{align*}7 \frac{2}{3}+\frac{4}{5}\end{align*}.
5. Divide \begin{align*}\frac{7}{8} \div -\frac{2}{3}\end{align*}.
6. Is the product of two rational numbers always a rational number? Explain your answer.

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