Equations:

The basis of mathematics is the equation.

The equation is a mathematical expression that contains the equal sign
What it means is that the left hand size of the equation (the part before the = sign) is equal to the right hand part of the equation (the part following the = sign).

A very simple example:

Well, that is fairly obvious, isn’t it? There is no way that six would not equal six.

In most equations, calculations are required. A simple example:

Where if you add 10 and 6 and then subtract 2, your result is 14. It cannot be 13, it cannot be 15, it cannot be 25. There is only one correct answer and that is 14.

Needless to say, equations will most likely involve calculations that will be more complex and complicated and most likely, will involve unknowns, which we will talk about in the next section. At the end of this section, we will provide you with some examples and a practice session.

The important thing to remember about equations is that whatever you do on one side of the equation, you have to do on the other side of the equation. To illustrate this on the preceding example:

Let us say for argument’s sake that we double the left side of the equation. If the above statement is correct, that means that we have to double the right side of the equation, too:

There you see that multiplying the left side by 2 requires multiplying the right side, as well.

So, you say, why is this important and how could it help? Let us consider another example.
Assume that you are asked to solve the following equation and calculate the value of x:

We are getting a little ahead of ourselves here, because we will only talk about the unknowns in the next section. However, we need an unknown here to show you what we mean. Here is the way you would calculate the above equation:

What we really want is to express the equation in such a way that all that we will be left with on the left hand side will be x and all the calculations will be on the right hand side, so that we will then be able to calculate the value of x. Of course expressing the equation in such a way has to be performed in an acceptable way.
First, we will multiply both sides of the equation by 2:

The part of the equation (X x 6) - 4 is now being both divided and multiplied by 2. Therefore, we can express the equation as:

Now we will add 4 to both sides of the equation:

Now we can say:

Now we will divide both sides of the equation by 6:

And our solution will be:

Now look at the examples provided and go through the practice session:

Equations with one unknown

Now that we have explored the concept of the equations and learned how we can handle the unknowns, let us show you a practical meaning of combining the two concepts:

The total of three consecutive numbers is 84. What are the numbers?
We know that there are three numbers, with the second one being higher than the first one by one and the third one being higher than the second one by one and they total 84.
Let us call the first number X.
Since the numbers are consecutive. the second number is higher by one and we can therefore call it: X + 1
The third number is going to be higher by one than the second number and we can therefore call it : X + 2
We know that the three numbers total 84, therefore we can express this in an equation:

To solve the equation is simple:

Since our first number is 27, our second number is going to be 27 + 1 = 28 and our third number is going to be 27 + 2 = 29. To perform a check:

Equations are a practical method of solving complex problems.

Equations with two unknowns

The real fun in mathematics starts when you can solve equations with two unknowns. Before you go on to equations with two unknowns, you have to be really comfortable solving equations with only one unknown and fully understand the concept of the unknowns. We will show you here what the concept is all about, but we would like to have you practice for a long (and we mean very long) time problems from the preceding section, before you go on to this level of difficulty problem and become discouraged.

Two sisters have a combined age of 25 years (If you add the age of one to the age of the other, the total is 25). If one is three years older than the other, how old is the older one?

This problem tells us two different things about the age of the sisters (and that is always what we need to know in a problem involving two unknowns - two different statements). One is that the ages of the two sisters added is 25 and the other is the difference in their age of 3 years. The secret is in formulating the information into two separate equations and then combining them into one:

Let us call the older sister O and the younger sister Y. Since we know that their ages add up to 25, we know that:

Since we know that the older is 3 years than the younger:

You cannot solve one equation with two unknowns. However if you have two, you can use the “definition” of one of the two unknowns from one of the two equations and substitute it into the second equation. Let us show you how it works. In the above example, the second equation defining the age difference between the two sisters tells us that:

We can use this “definition” of the O unknown (Y + 3) and substitute it into our other equation:

Let us see what will happen:

We have substituted (Y + 3) for O in our second equation and now we no longer have an equation containing two unknowns, only one and we can easily solve that:

Substituting the value of 11 for Y in either one of our original equations:

Therefore our answer is that the sisters are 11 and 14 years old and the older sister is 14 years old. Let us see, whether the answer fits the definition of the problem:

The above method is called the Substitution method, since you are substituting the value of one of the variables from one of the equations into the other equation. Another method is called the Elimination method and here is an example:

In the Elimination method, you will pick one of the two variables (X or Y) and ensure that you have the same number of the variables, in both equations.
The two variables MUST have different signs!
In the above example, we will multiply the entire 2nd equation by 2, so that we will have 2X in both equations:

However, we must have the sign before 2X changed to minus, we will therefore multiply the entire equation by -1:

We will now write the two equations as follows:

At this point we will subtract 2X from 2X, subtract 4 from -3Y and subtract 20 from 6, resulting in:

and since:

The Elimination method is sometimes faster than the Substitution method, however we will be using the Substitution in all our examples in our problems.