# Equations- A Basic Introduction

Applied mathematics is one which is used in day-to-day life, in solving troubles (problems) or in business purposes. Let me write an example:

George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he?

If you are familiar with day-to-day calculations – you must say that George had 41 dollars, and since he had 41, gave 14 to Matthew saving 27 dollars. That’s right? Off course! This is a general (layman) approach. ‘How will we achieve it mathematically?’ –we shall restate the above problem as another statement (meaning the same):

George had some money $x$ dollars. He gave 14 dollars to Matthew. Now he has 27 dollars. How much money he had? Find the value of $x$ .

This is equivalent to the problem asked above. I have just replaced ‘some money’ by ‘x dollars’. As ‘some’ senses as unknown quantity— $x$ does the same. Now all we need to get the value of x.
When solving for $x$ , we should have a plan like this:

 George had $x$ dollars. He gave to Matthew 14 dollars Now he must have $x-14$ dollars

But problem says that he has 27 dollars left. This implies that $x-14$ dollars are equal to 27 dollars.
i.e., $x-14=27$

$x-14=27$ contains an alphabet x which we assumed to be unknown–can have any certain value. Statements (like $x-14=27$ ) containing unknown quantities and an equality are called Equations. The unknown quantities used in equations are called variables, usually represented by bottom letters in English alphabet (e.g.,$x,y,z$ ). Top letters of alphabet ($a,b,c,d$ ..) are usually used to represent constants (one whose value is known, but not shown).

Now let we concentrate on the problem again. We have the equation x-14=27.

Now adding 14 to both sides of the equal sign:
$$x-14 +14 =27 +14$$
or, $x-0 = 41$ as (-14+14=0)
or, $x= 41$ .
So, $x$ is 41. This means George had 41 dollars. And this answer is equal to the answer we found practically. Solving problems practically are not always possible, specially when complicated problems encountered —we use theory of equations.

To solve such equations, you need to know only four basic operations viz., Addition, Subtraction, Multiplication and Division; and also about the properties of equality sign.

We could also deal above problem as this way:
$x-14= 27$
or,$x= 27+14 =41$
-14 transfers to another side, which makes the change in sign of the value, i.e., +14.

When we transport a number from left side to right of the equal sign, the sign of the number changes and vice-versa. As here -14 converts into +14; +18 converts into -18 in example below:
$x+18 =32$
or, $x=32 -18 =14$ .

Please note, any number not having a sign before its value is deemed to be positive- -e.g., 179 and +179 are the same, for the sake of theory in equations.

Before we proceed, why not take another example?

Marry had seven sheep. Marry’s uncle gifted her some more sheep. She has eighteen sheep now. How many sheep did her uncle gift?

First of all, how would you state it as an equation?
$7 + x = 18$
or, $+7 +x =18$ (just to illustrate that 7=+7)
or, $x= 18-7 =9$ .
So, Marry’s uncle gifted her 9 sheep. ///

Now tackle this problem,

Monty had some cricket balls. Graham had double number of balls as compared to Monty. Adam had also 6 cricket balls. They all collected their balls and found that total number of cricket balls was 27. How many balls had Monty and Graham?

As usual our first step to solve this problem must be to restate it as an equation. We do it like this:
Monty had (let) x balls.
Then Graham must had $x \times 2=2x$ balls.
Adam had 6 balls.
The total sum=$x+2x+6=3x+6$
But that is 27 according to our question.
Hence, $3x+6=27$
or, $3x=27-6 =21$
or,$x=21 /3 =7$ .
Here multiplication sign converts into division sign, when transferred.

Since $x=7$ , we can say that Monty had 7 balls (instead of x balls) and Graham had 14 (instead of $2x$ ).

# Types of Equations

They are many types of algebraic equations (we suffix ‘Algebraic’ because it includes variables which are part of algebra) depending on their properties. In common we classify them into two main parts:

1. Equations with one variable (univariable algebraic equations, or just Univariables)
2. Equations with more than one variables (multivariable algebraic equations, or just Multivariables)

## Univariable Equations

Equations consisting of only one variable are called univariable equations.

All of the equations we solved above are univariables since they contain only one variable (x).

Other examples are:
$3x+2=5x-3$ ;
$x^2+5x +3=0$ ;
$e^x =x^e$ (here, e is a constant).

Univariables are further divided into many categories depending upon the degree of the variable. Some most common are:

1. Linear Univariables: Equations having the maximum power (degree) of the variable 1.
$ax+b=c$ is a general example of linear equations in one variable, where a, b and c are arbitrary constants.
2. Quadratic Equations: Also known as Square Equations, are ones in which the maximum power of the variable is 2.
$ax^2+bx+c=0$ is a general example of quadratic equations, where a,b,c are constants.
3. Cubic Equations: Equations of third degree (maximum power=3) are called Cubic.
A cubic equation is of type $ax^3+bx^2+cx+d=0$ ; where a,b,c,d are constants.
4. Quartic Equations: Equations of fourth degree are Quartic.
A quartic equation is of type $ax^4+bx^3+cx^2+dx+e=0$ .

Similarly, equation of an n-th degree can be defined if the variable of the equation has maximum power n.

## Multivariable Equations

Some equations have more than one variables, as $ax^2+2hxy+by^2=0$ etc. Such equations are termed as Multivariable Equations. Depending on the number of variables present in the equations, multivariable equations can be classified as:

1. Bi-variable Equations – Equations having exactly two variables are called bi-variables.
$x+y=5$ ; $x^2+y^2=4$ ; $r^2+\theta^2=k^2$ , where k is constant; etc are equations with two variables.
Bivariable equations can also be divided into many categories, as same as univariables were.

A. Linear Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 1.
For example: $ax+by=c$ is a linear but $axy=b$ is not.
B. Second Order Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 2.
For example: $axy=b$ , $ax^2+by^2+cxy+dx+ey+f=0$ are of second order.
Similarly you can easily define n-th order Bivariable equations.

2. Tri-variable Equations: Equations having exactly three variables are called tri-variable equations.
$x+y+z=5$ ; $x^2+y^2-z^2=4$ ; $r^3+\theta^3+\phi^3=k^3$ , where k is constant; etc are trivariables. (Further classification of Trivariables are not necessary, but I hope that you can divide them into more categories as we did above.)
similarly, you can easily define any n-variable equation as an equation in which the number of variables is n.

Out of these equations, we have discussed only Linear Univariable Equations here.

We have already discussed them above, for particular examples. Here we’ll discuss them for general cases.

As told earlier, a general example of linear univariable equation is $ax+b=c$ .

We can adjust it by transferring constants to one side and keeping variable to other.
$ax+b = c$
or, $ax = c-b$
or, $x = \frac{c-b}{a}$
this is the required solution.

Example: Solve $3x+5=0$ .
We have, $3x+5=0$
or, $3x = 0-5 =-5$
or, $x = \frac{-5}{3}$

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