In Economics, a line representing supply and a line representing demand are often graphed on the same coordinate plane, and the lines intersect at one point. How many solutions does this system of linear equations have? Is the system consistent, consistent-dependent, or inconsistent?

### Consistent and Inconsistent Linear Systems

Solutions to a system can have several forms:

- One solution
- Two or more solutions
- No solutions
- An infinite number of solutions

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

#### Inconsistent Systems

A system with parallel lines will have **no solutions**.

Remember 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, so they have no solution.

A system with no solutions is called an **inconsistent system. **

#### Let's solve the following system of equations:

\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.”

This is an inconsistent system.

#### Consistent Systems

**Consistent systems,** on the other hand, 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 is commonly practiced in linear systems
- 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 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**.

#### Let's solve the following system of equations:

\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*}

Add the equations together.

\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.

The lines are coincident and therefore there are an infinite number of solutions. This is a consistent-dependent system.

#### Now, let's identify the following systems as consistent, inconsistent, or consistent-dependent:

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

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:

- These lines are parallel.
- The system has no solution.
- The system is inconsistent.

#### Finally, for the following problem, let's determine if the system is consistent or inconsistent:

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?

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.

### Examples

#### Example 1

Earlier, you were told that a line representing supply and a line representing demand are often graphed on the same coordinate plane in Economics and that the lines intersect in one point. How many solutions does this system of linear equations have? Is the system consistent, consistent-dependent, or inconsistent?

Since the lines intersect in a single point, the lines have one solution. There is at least one solution so the system is not inconsistent. This means that the system is either consistent or consistent-dependent. It is not consistent-dependent because the lines are not the same and do not have an infinite number of solutions. Therefore, the system is just consistent.

#### Example 2

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?

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.

### Review

- Define an inconsistent system. What is true about these systems?
- What are the three types of consistent systems?
- You graph a system and see only one line. What can you conclude?
- You graph a system and see the lines have an intersection point. What can you conclude?
- The lines you graphed appear parallel. How can you verify the system will have no solution?
- 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.

- \begin{align*}&3x-4y=13\\ & y=-3x-7\end{align*}
- \begin{align*}&4x+y=3\\ &12x+3y=9\end{align*}
- \begin{align*}&10x-3y=3\\ &2x+y=9\end{align*}
- \begin{align*}&2x-5y=2\\ &4x+y=5\end{align*}
- \begin{align*}&\frac{3x}{5}+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+2y=4\\ &-2x+2y=24\end{align*}
- \begin{align*}&5x-2y=3\\ &2x-3y=10\end{align*}
- \begin{align*}&3x-4y=13\\ & y=-3x-y\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{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*}
- 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 cost?
- 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?
- 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?
- 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?
- 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, 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**

- 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?
- Find an equation for the line perpendicular to \begin{align*}y=-\frac{3}{5} x-8.5\end{align*} containing the point (2, 7).
- Rewrite in standard form: \begin{align*}y=\frac{1}{6} x-4\end{align*}.
- Find the sum: \begin{align*}7 \frac{2}{3}+\frac{4}{5}\end{align*}.
- Divide: \begin{align*}\frac{7}{8} \div -\frac{2}{3}\end{align*}.
- Is the product of two rational numbers always a rational number? Explain your answer.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.6.