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.