Types of Solutions to Systems of Linear Equations

In this post, we will learn that any system of linear equations has one of three possible types of solution. Each type of solution has distinctive characteristics. These will be presented, so we can understand the type of system or situation we are dealing with when we observe a system of linear equations. The three types of solution are:

1. One unique solution
2. Infinitely many solutions
3. No solution

The type of solution can be determined by characteristics of lines used in analytic geometry. Those are the slope, the y-intercept, and the graph of the lines themselves. These are discussed in more detail below. However, it is important to understand that a system of equations has NO solution when the number of unknowns exceeds the number of equations. Therefore, a linear system that has 2 equations and 3 unknowns cannot be solved. Moreover, systems of 2 linear equations with 2 unknowns will be used in the examples below.

One Unique Solution

A system of linear equations has a unique solution when the graph of the lines intersect at a single point. This point of intersection is the solution of the system. Based on this, it can be inferred that the lines have distinct slope and y-intercept. A system of linear equations with an unique solution is shown below.

The graph generated in a TI-89 Titanium calculator shows the intersection of the lines at point (1, 3). This is the solution of the system of equations.

Infinitely Many Solutions

A system of linear equations has an infinite number of solutions when the graph of the lines are the same. In other words, the equations are the same. They are just written differently. Since the equations are the same, they obviously have the same slope and the same y-intercept. A system of linear equations with an infinite number of solutions is shown below.

The graph generated in a TI-89 Titanium calculator seems to show a single line. However, it shows the graph of both equations. This is because the equations are the same and their graphs touch each other at every point, hence infinitely many solutions. In fact, if the second equation is divided by -3 on both sides, the first equation is obtained.

No Solution

It was discussed previously that systems of linear equations with more unknowns than equations have no solution. Additionally, a linear system has no solution when the graph of the lines are parallel. In other words, they have no intersection point. These lines have the same slope, but different y-intercept. A system of linear equations with no solution is shown below.

The graph generated in a TI-89 Titanium calculator shows two parallel lines. This means the lines will never intersect and the system has no solution.

Conclusion

When dealing with system of linear equations, no matter the size, they will have one of these three types of solution. 

Finding Composite Functions

A composite function is one that obtained by plugging in an inner function into an outer function. In other words, a function takes the place of the independent variable in an outer function. To understand this concept, consider two functions f and g. Then, the composite function is denoted by f ◦ g which is defined by:

In other words, the function g is plugged into the function f.

To understand how to find composite functions, we will look at a few examples. 

Example 1:

It is also possible to have a composite function of three or more functions. The same rules of the previous examples apply. An example is shown next.

Example 2:

Finding the Inverse of a Function

First of all, it is important to know that not all functions have an inverse. Only one-to-one functions have an inverse function. 

But what is a one-to-one function?

A one-to-one function is one that never takes the same value twice. For example, the function f(x) = x is one-to-one, because f(1) = 1, and no other value of x can have a value of f(x) that is equal to 1. 

On the other had, the function f(x) = x² is not one-to-one, because f(-2) = 4 and f(2) = 4. As noted, there are two values of x that take the value of 4.

Now back to the inverse of a function:

The inverse of a function f(x) is denoted by f^-1(x). It is important to understand that -1 is not an exponent, but just notation. It denotes that it is the inverse function of f(x). Now, we will look at a couple of examples to learn how to calculate the inverse of a function.

Example 1:

Example 2:

The steps explained in the examples can be used as an algorithm to find the inverse of a function (when possible, of course).

Basic Rules of Differentiation (Part 3)

In Part 1 and Part 2, we learned the rules of differentiation to handle common functions we might encounter. Now, we will learn about chain rule which will close this chapter of the basic rules of differentiation.

Chain Rule:

Chain rule is perhaps the most powerful of all rules of differentiation. This is due to the fact that many other rules of differentiation are based off it. In order to understand this rule, consider a function f(x) that is a composite function of the functions g(x) and h(x). In this case, the function h(x) is inside the function g(x).

It can be observed that g(x) is the outside function, and h(x) is the inside function. Then, the derivative of f(x) is given by:

This is known as chain rule. It basically says the derivative of f(x) is "the derivative of the outside (keep the inside function the same) times the derivative of the inside."

Examples:

Let's observe some examples of how we can apply chain rule. It is important to understand that the other rules of differentiation still apply.

Once we know how to apply chain rule, we can say that we know how to differentiate. Chain rule is the "pillar" of differentiation. If we understand it, we will be able to learn more complex rules of differentiation and even integration. 

Basic Rules of Differentiation (Part 2)

In Part 1, we learned two basic rules that allow us to differentiate polynomial functions. Now we will learn to deal with functions that involve the multiplication or division of functions. We will learn about product rule and quotient rule.

Product Rule:

Suppose we have a function f(x) that is the result of the multiplication (known as a product) between two functions g(x) and h(x). Then, the derivative of f(x) is given by:

Many teachers and students play with words to memorize this rule. Anyone can make up his/her own way to remember it, but it basically says the derivative of f(x) is "the first times derivative of the last, plus the last times derivative of the first."

Quotient Rule:

Suppose we have a function f(x) that is the result of the division (known as a quotient) of the functions g(x) by h(x). Then, the derivative of f(x) is given by:

In this case, we can say the derivative of f(x) is "the bottom times derivative of the top, minus the top times derivative of the bottom. All divided by the bottom squared."

Examples:

These two rules are quite popular, so let's look at some examples. Notice that we will apply the rules learned in Part 1 when dealing with the terms of the functions.

Now, as seen in the examples, we can differentiate more complicated functions!!!

Graphing a Simple Linear Function

In this post, we present an example of how to graph a linear function in the xy-plane from scratch.

Example

Graph the line described by the function y = -5x + 1.

Solution:

The function y = -5x + 1 is of the form y = mx + b. This indicates the graph is a straight line. For this reason, we can graph the line with only two points that we can connect to form the line.

Select any values of x and plug them into the given function to obtain the corresponding values of y. For this example, we use the values x = -1 and x = 1. 

The table below summarizes the results above.

Trace a line connecting these two points to obtain the graph for the function y = -5x + 1.

The Equation of the Line in Form y = mx + b

Definition

The equation of the line in form y = mx +b represents the basic relationship between two variables (x and y). m is the slope which is a measure of the inclination of the line. b is the y-intercept which is the value of y when x = 0.

The simplest of these functions is the equation y = x. This is a line with slope m = 1, and the y-intercept is b = 0. The graph for the function y = x is shown below.

As we can observe, the graph of y = x is an inclined line in the xy-plane.

The Slope

Mathematically, the slope can be described as the change in the dependent variable (y) over the change in the independent variable (x). This is given as:

This is true for any two points on the line. In other words, in order to calculate the slope, we need to know at least two points on the line.

The y-Intercept

As stated above, the y-intercept is the value of y when x = 0. This can be observed directly if the graph is known, or obtained mathematically from a point on the line (provided the slope was previously found).

Example

Write the equation in form y = mx + b for the line passing through the points (-2,-10) and (1,2).

Solution:

ALWAYS start with the slope. In this case, we can use the given points and plug them into the formula for the slope. Consider Point 1 as (-2,-10) and Point 2 as (1,2).

The slope is found to be 4. Therefore, the equation so far is y = 4x + b. We still need to find b.

In order to find b, we can use any of the two given points into what we have for the equation so far.

For simplicity, we can use (1,2).

The equation of the line passing through the points (-2,-10) and (1,2) is:

The graph of the line is:

From the line in the graph, it can be observed that y increases by 4 units as x increases by 1 unit. In addition, we can see that the y-intercept is -2.