In the previous section, we looked at the first graph of a mixed-degree system. The solution is shown by the intersection of the lines on the graph of equations. But if the lines do not intersect, a solution cannot be found. The graphs of an equation must be parallel.

## Figure 2

In Figure 2, a mixed-degree system has exactly one solution, and the solution has the same x-value as the initial x-value. This is called Euler’s rule. The rule applies even when there are odd numbers of bridges connecting two different landmasses.

When the number of firms in a market is small, a long-run equilibrium is reached. This occurs when a typical firm produces output Q1 at a price P1, where price equals the average total cost. The equilibrium is long-run, and profits are maximized at the monopolistically competitive price.

## Figure 4

A mixed-degree system is a system with one or more solutions. For example, consider a long-distance phone service. Originally, long-distance phone service was a monopoly. As the price of tap water increases, demand for bottled water increases. This result in a shift of the demand curve to the right. The profit-maximizing price is P2, and is shown on the demand curve.

## Figure 6

A mixed-degree system with exactly one solution is a system that has exactly one solution and two solutions that are not the same. In this example, the problem is to find the point M that lies in the boundary of the coordinate plane. The boundary line is drawn as a dashed line. Points M and N give true statements.

## Figure 7

A solution to a system of equations is a value that is true for all the equations in the system. This value can be one of three outcomes. One way to know how many solutions there are to a system is to plot the equations against one another. In Figure 7 below, we see two lines that make up a system. The lines are parallel if they do not intersect.

## Figure 8

In mathematics, a solution for a system of equations is a value or values that are true for all the equations in the system. For example, a system of equations with two variables and two solutions has a solution if the two lines intersect, and vice versa. If the two lines are parallel, there is an infinite number of solutions.

## Figure 9

A mixed-degree system is a system where the chemical components of the system are not the same as those of the opposite solution. Alcohol molecules, for example, contain three carbon atoms and one OH group. The intermolecular force between these molecules is what determines their physical properties. Among these properties are molar mass and boiling point.

## Figure 10

An equilibrium in a mixed-degree system is a system in which the supply and demand curves intersect at a point. This point is called a point M. Each stand produces a certain quantity q1 of pretzels. Thus, the total number of pretzels produced equals 1,000 x q1, and each stand makes a zero profit. The price, therefore, equals the average cost of production.

## Figure 11

A mixed-degree system with exactly one solution is shown in Figure 11. Historically, long-distance phone service has been a monopoly. In the long run, the market returns to equilibrium when entry into an industry drives down prices to the minimum average total cost. The average total cost is the price at which a typical firm can produce Q1.

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