Learning about Different Algebraic Equations and How to Solve Them

There are only a handful of people who can confidently say that they love solving math problems. But the number would have been different if people could see how interesting it was to solve algebraic equations. If you are willing to give it a try, you should start by learning about the different algebraic equations and how to solve them in the simplest manner.

There are plenty of online algebra calculator that do a decent job at solving various algebraic equations almost instantly. However, if you really want to learn about the equation-solving processes, you need to practice it without the help of such tools. In this blog, we will discuss several types of algebraic equations and how you can solve them.

What Are Algebraic Equations?

An algebraic equation is a mathematical statement where two expressions are set equal to each other. Even though the term equation is self-explanatory, let’s simplify it for you.

The purpose of an equation is to equate one quantity with another. They are quite similar to a balance scale. When you put an equal amount of weights on both sides of the scale, then it is considered balanced. An equation follows the same logic. The one side of the equation must have the exact same value on the other side. Otherwise, it becomes an inequality.

Types of Algebraic Equations

There are various types of algebraic equations that you, as a math student, need to learn. Some of the most common equations in algebra are:

Polynomial Equations:

Polynomial equations consist of variable terms along with whole-number exponents. Usually, these algebraic equations are classified by the number of terms in the expression. For instance, if there are two terms in the expression, it will be called binomial. For having more than three terms in the expression, it will be called trinomial. In simpler words, any expression with more than one term is called a polynomial.

Polynomial equations are also classified by degree, which is the number of the highest exponent in the given expression. Polynomial equations with degree one are called linear, while the equations with degrees two and three are called quadratic and cubic polynomials, respectively.

Exponential Equations:

Exponential equations are those which have variable terms in the exponents. Here is what an exponential equation looks like this –

Y = 4^(x-3) + 6

If the independent variable in the equation has a positive coefficient, its exponential function is exponential growth. On the other hand, the exponential function is classified as exponential decay if it has a negative coefficient. The exponential growth equations are used to describe financial concepts like compound interest and the spread of populations and diseases. Exponential decay equations are often used for describing phenomena like radioactive decay.

Rational Equations:

This type of algebraic equations comes in the form p(x) / q(x), where both p(x) and q(x) are polynomials. Here is an example of a rational equation –

(x – 5) / (x^2 – 3x + 7)

One of the most interesting facts about rational equations is that they have asymptotes, which are the values of x and y that the graph of the equation approaches but is never able to reacḥ. If you explore this segment deeper, you will learn about vertical asymptotes of a ration equation. Such asymptote is an x-value that the graph never reaches —  the y value either goes to negative or positive infinity as the value of x approaches the asymptote. There is also a horizontal asymptote which is a y-value that the graph approaches when x goes to negative or positive infinity.

Trigonometric Equations:

It is easy to guess that trigonometric equations are those which contain the trigonometric functions – sin, cos, tan, sec, cosec and cot. Trigonometric functions describe the ratio between two arms of a right triangle while taking the angle measure as the input or independent variable and the ratio as the output or dependent variable.

Trigonometric functions are unique in that they are periodic. It means the graph repeats after a certain period of time. For your information, the graph of a standard sine wave has a period of 360 degrees.

These were some of the common types of algebraic equations. Now, let’s focus on some techniques that will help you solve various algebraic equations.

How to Solve Algebraic Equations?

Solving algebraic equations may not seem easy at first, so let’s talk about the bit that may appear comparatively easier for first-time learners of algebra.

Solving a linear equation:

If the linear equation has only has one variable, you can solve it in the following manner.

x + 3 = 7

x + 3 – 3 = 7 – 3

x = 4

Now, when there are more than one variables, you are generally are provided with two equations consisting of the same set of variables. Let’s see how you can solve such equations.

2x + y = 11, 3x – y = 4

Let’s add 3x – y on both sides of the first equation.

2x + y + 3x – y = 11 + 3x – y

Now, we know that 3x – y = 4. So, replacing the 3x – y on the right hand side of the equation, we get:

2x + y + 3x – y = 11 + 4

5x = 15

5x / 5 = 15 / 5

x = 3

Now, that we know the value of x, let’s put the value in one of the equations to find out the value of y.

2x + y = 11

2*3 + y = 11

6 + y = 11

6 + y – 6 = 11 – 6

Y = 5

Solving a quadratic equation:

There are a number of ways in which you can solve a quadratic equation.

1. Factoring
2. Completing the square
3. Using quadratic formula
4. Graphing

Since we are looking for the simplest way of solving the algebraic equations, let’s just focus on completing the square method.

Let’s say, you are given the equation, ax² + bx + c = 0

The first step would be to divide all the terms by a (the coefficient of x²).

x² + (b/a)x + c/a = 0

The next step is to move the number term (c/a) to the right side of the equation.

x² + (b/a)x = – (c/a)

Now, you need to complete the square on the left hand side of the equation and balance the equation by making an addition of the same value to the right hand side of the equation.

We will get something that will look like (x + p)² = q, which can be solved quite easily.

Take the square root on both sides of the equation,

Next, subtract the number that remains on the left of the equation to find x.

Well, there are more to learn about algebraic equations and how to solve even more complex equations. But that is for the next time. Now, you can simply refer to these basic details to get started with your algebra lessons.

Author bio: steven parker is a math teacher at a high school in Hobart, Australia. He is also a part of the team of experts at MyAssignmenthelp.com. He has even been one of the supervisors in the development of the online algebra calculator for the website.