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. 

Solving Simple Systems of Linear Equations (Part 3)

Previously, the methods of elimination of variables and substitution of variables were introduced (Part 1 - click here). Subsequently, these methods were used to solve a couple of linear systems of 2 equations with 2 unknowns (Part 2 - click here). Now in this post, we take it a little further. We will find the solution to a system of 3 linear equations with 3 unknowns. 

Example:

The methods of elimination and substitution of variables can be applied to all kinds of systems of linear equations, provided they have a solution. However, as these systems become larger in number of equations and unknowns, these methods become cumbersome. Instead one can use other methods to handle large systems of linear equations such as the application of matrices or even numerical methods. These will be discussed in future posts.

Solving Simple Systems of Linear Equations (Part 2)

This post is a continuation of the previous one (click here) which introduced two methods of solving systems of linear equations. We will apply either of these two methods to show how to use them to solve systems of linear equations.

Example 1:

Example 2:

The methods of elimination of variables and substitution of variables were applied to solve the above examples, respectively. You are encouraged to use the other method to solve each example. Since the solutions are already known, you can confirm your results.

Solving Simple Systems of Linear Equations

What is a linear equation?

An algebraic equation is considered a linear equation simply because its graph is a straight line. You can check the post about graphing a simple linear function in the link that appears in parentheses (click here). In addition, a linear equation is one in which the variables involved have an exponent of 1. We can find a few examples of linear equations below.

As observed in the equations above, all the variables have 1 as an exponent. This makes them linear equations. The first two equations are linear equations with two unknowns, while the third equation is a linear equation with three unknowns. We will focus on the former in the remainder of this article.

What is a system of linear equations?

A system of linear equations is one that involves two or more linear equations with common variables as unknowns. When dealing with systems of linear equation, we are usually asked to find the solution of the system. The solution of the system is the value of the variables. 

In this article, we will only deal with systems of two linear equations with two unknowns. These are the simplest types of systems. When solving them, we often only apply two different methods. We will discuss these next.

Elimination of Variables

In this method, we seek to combine the given equations in such a way that we can eliminate one of the variables. From the combination of the equations, we can obtain a single equation with a single unknown that can be found by standard mathematical procedures. Subsequently, we will follow an example in which we apply this method.

Example 1:

Substitution of Variables

In this method, we solve either of the given equation for one of the unknowns. Then, the chosen equation is plugged into the other equation. Again, we obtain a single equation with a single unknown that can be found easily. 

Example 2:

Conclusion

As observed, the same solution was obtained by using the two different methods. There are more methods to solve linear equation. However, you will only need these two when solving simple systems of two linear equations with two unknowns. Needless to say, either method can applied. Your choice will depend on the type of system you have at hand. As you become more proficient, you will be able to choose the best for each particular system.

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:

Introduction

Modern society and living conditions are made possible by advances in science, technology, engineering and mathematics (STEM). The subjects studied in these fields have become so relevant that any person not familiar with their basic principles is at a disadvantage. Consequently, the study of STEM fields must be encouraged throughout academic education. Even then, no one should stop there. Real things are dynamic, always changing, so one must engage in lifelong learning. In fact, this is a personal motivation to start this blog.

So what lies ahead?

During many years in higher education, I have learned that complex problems can be solved by breaking them down into simpler problems. In STEM fields, this means analyzing the problems from the fundamental concepts. In other cases, students might also need a reference in topics that are of great importance. As a result, the entries in this blog will present these concepts with thorough explanations that can help any student at different academic levels.

Statement of Purpose

Provide students of all ages and levels with articles and information that can help them understand a variety of concepts of STEM fields. I will do my best to organize and label entries with key words, so anyone can easily find what they look for.

Anyone reading the blog is welcomed to comment, make suggestions and point out errors. I hope this becomes helpful to many around the world.

Enjoy!

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).

Trigonometric Functions

In order to understand trigonometric functions, we can consider the right triangle with an angle θ and the corresponding sides as shown below. The opposite side is the side in front of the angle being observed; the adjacent side is the side touching the vertex where the angle being observed is located; and the hypotenuse is the longest side of the triangle.

For simplicity, we will use the first letter of each side to show the trigonometric functions. The basic trigonometric are cosine, sine and tangent along with their respective reciprocals secant, cosecant and cotangent. These are defined below.

As stated previously, secant, cosecant and cotangent are the reciprocals of cosine, sine and tangent, respectively. Therefore, they can also be written as:

Now that we are familiar with the basic trigonometric functions, we can proceed to analyze right triangles and other type of triangles as we will learn in the future.

Stoichiometry in Combustion (Part 2 - Example with Excess Air)

Determine the air-fuel ratio on a mass basis for the complete combustion of octane, C8H18, with 150% theoretical air (50% excess air).

Solution:

The first part of the solution is to obtain the equation for the complete combustion of octane with the theoretical amount of air. Find the coefficients of the equation.

The second part of the solution is using the left hand side of the chemical equation to obtain the new coefficients of the products when 150% theoretical air is used. Now, the air is multiplied by 1.5, and there is excess air that appears as uncombined oxygen in the products.

The third part of the solution is actually calculating the air-fuel ratio which is what the problem asked for.

The ratio of the amount of air in this reaction to the amount of fuel is about 22.7.

Bibliography:

Michael J. Moran, Howard N. Shapiro. Fundamentals of Engineering Thermodynamics. John Wiley & Sons, Inc., 2008.

Stoichiometry in Combustion

Stoichiometry is the study of the relationship between quantities of reactants and products in chemical equations. It is based on the principle of conservation of mass. Therefore, it involves balancing both sides of the equation by assuring there are the same number of atoms for all elements involved. 

In the field of mechanical engineering, stoichiometry is mostly used in the field of thermodynamics when studying combustion reactions. In this type of reactions, a fuel reacts chemically with oxygen in order to release energy and combustion products. When a combustion is complete, all the carbon present in the fuel is burned to carbon dioxide, all the hydrogen is burned to water, all the sulfur is burned to sulfur dioxide, and all other combustible elements are fully oxidize. If these conditions are not met, the combustion is said to be incomplete. Now, we will observe a couple of examples.

Example 1:

Balance the equation below for the complete combustion of methane.

Solution:

By applying the principle of conservation of mass, we can create equations to balance the number of atoms of the same elements on both sides in terms of the coefficients of each compound in the reaction.

By solving the system of four equations and four unknowns, the balanced chemical equation is:

Example 2:

For the balanced equation in the previous example, obtain the air-fuel ratio.

Solution:

The ratio of the amount of air in this reaction to the amount of fuel is about 17.24.

Bibliography:

Michael J. Moran, Howard N. Shapiro. Fundamentals of Engineering Thermodynamics. John Wiley & Sons, Inc., 2008.

Use of Pythagorean Theorem

One of the most important formulas in math is Pythagorean theorem which is named after Greek philosopher Pythagoras (even though its formulation is historically unknown). The importance of Pythagorean theorem lies in the fact that it can be used to find the length of the side of a right triangle, provided the length of the two other sides is known. It is given by:

Where c is the longest side  of the triangle (also known as hypotenuse), and a and b are the two other sides in no particular order.  The right triangle with its corresponding sides is shown in the figure below.

Now we can take a look at a simple example of how we can use Pythagorean theorem to find the length of a side.

Example:

Find the length of the hypotenuse of a right triangle in which the two other side have lengths of 3 and 4 units, accordingly.

Solution:

We can sketch the right triangle described in the problem statement. The triangle below is not drawn to scale.

By using Pythagorean theorem, we can calculate the length of the hypotenuse as shown below.

The length of the hypotenuse of this right triangle is 5 units.

Closing Remarks:

Right angles and triangles are very common. They appear all around us, thus the applications of Pythagorean theorem are endless. This makes it a very powerful formula which must be dominated by any math student. We will see it over and over again!

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!!!

Basic Rules of Differentiation (Part 1)

There are two basic rules that we can consider the "baby steps" to begin learning calculus. These are the derivative of a constant and the derivative of a variable raised to a constant exponent. 

Derivative of Constant:

The derivative of any constant is zero. Consider the following function f(x), where c is any constant.

Derivative of a Variable Raised to a Constant Exponent:

This is the second rule of differentiation that any calculus student must learn. Consider the following function f(x) where a is any real valued exponent.

In this case, we basically bring down the exponent to the front, and then, subtract 1 to the exponent.

Examples:

Now we will have a few examples that show how important these two rules are. Basically, the derivative of any polynomial function can be obtained from just these two rules.

Many real life situations can be described by polynomial functions, so it is safe to say that these two rules of differentiation will be used frequently.

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.