[Maths Class Notes] on Variable Pdf for Exam

What is a Variable?

In algebra, there are many occurrences of unknown values depicted by symbols or letters. These are variables. Today, we take a look at what are variables and its uses in different areas like algebra and statistics. To begin, variables are letters such as x, y, z, r, or call it any letter you desire, which is used to represent an unknown number. A variable is a quantity that may change within the context of a mathematical experiment or problem. Usually, a single letter is used to represent a variable. The letter x, y, and z are the generic symbols used for the variables. Sometimes, we may also choose a letter that reminds us of the quantity it represents, such as t for time, b for bacteria, or v for voltage.

Usually, when we do not know the value of a certain number, we use a variable until we figure out what the actual number is. The standard variables that we often use are x and y, with x being the most popular. Once you know the value of the variable, you can replace the variable with the calculated value.

Why are Variables Used?

Often in mathematical problems, you will discover equations with variables like x – 3 = 7. Here, you must understand that variables do not just exist for algebra quizzes but have a practical purpose.

Suppose you are at your favourite burger place. They sell cheeseburgers for Rs. 4 each. You want to get as many possibilities as you can, but you only have Rs. 13. Along with buying many cheeseburgers, you will also want a drink that costs Rs. 1. Now, how will you figure out how many burgers you want? Here, the number of burgers is your variable; maybe it’s 1 or 3 you do not know. So, let’s call it x.

Now, each burger costs Rs. 4, so 4x is the cost of the burger multiplied by the number of burgers. 4x + 1 is the cost of the burger and the drink which equals to Rs. 13 the money you have. Therefore, our equation comes to 4x +1 = 13.

You have just used a variable to help you solve a critical problem and help you maximize the cheeseburger intake. 

Let us solve this equation to find out how many burgers you can get

4x + 1 = 13

4x = 13 -1

4x = 12

Now, x = 12 /4 = 3 

So, you can now have 3 burgers that cost Rs. 4 each along with a drink of Rs. 1, which equals to Rs. 13, the money you have with you.

Solved Examples of Variables

Example 1 

8x = 64

8 (multiplied by) x = 64

Here, x is the number that you should multiply by 8 to get a product of 64.

Solution: By using the table of multiplication we know that the number is 8 for x

8 (multiplied by) 8 = 64

Alternatively, we can isolate the variable to one side (on the LHS)

So, 8x=64 can be written as x = 64/8 i.e. x = 8

When we solve an equation, we figure out what value of x or any other variable makes the equation true. 

Example 2

Which number will be the solution to the equation (14 – x = 7)? 

Is x 2, 7, or 8?

Solution: Here, you are provided with the number 2, 7, and 8, and only one of these number satisfy the equation. If you do not know the exact solution, you must investigate which of the given number will give the correct value to the equation and will represent the variable x 

Let us find by using all the numbers given

When, x = 2 then, 14 – 2 = 12 (wrong equation value)

When, x = 7 then, 14 – 7 = 7 (correct equation value)

When, x = 8 then, 14 – 8 = 6 (wrong equation value)

Therefore, the correct number in place of x is 7

Some other variable equation examples are listed below

  1. 3x – 4 = 11

  2. 2x +1 = 7

  3. y + 6 = 10

  4. 6x – 3 = 12

Here, x and y are known as variables.

What are Constants and Coefficients?

Now, let us know what the other things around a variable are. Let us for example take an equation 3x + 4 = 5

Here, 4 and 5 are numbers but are also known as constant. A constant cannot be changed, i.e., they are the fixed quantity. A number can never be used as a variable as they are constant. 

Looking further at the equation, we also have number 3, and it is known as a coefficient. So, the coefficient is a number that multiplies a variable; for instance, if you have 6y than the coefficient is 6 if you figure out that value of y is 5, then 6y is 6×5, which is 30.

Variables in Statistics

Statistics uses variables in terms of real-life scenarios. It can also be considered as an attribute here. It is used to represent people, places, things, etc. For example, income is a variable that can vary between different people. It is any number or characteristic that can be measured or counted. It is sometimes also called as a data item in statistics. Examples of variables in statistics include age, gender, race, class grades, income, car type or laptop manufacturing year. 

Types of Variables

In maths there are two types of variables:

  1. Dependent variable

  2. Independent variable

Dependent Variables

It is characterized as a variable whose quality depends on the estimation of another variable in its condition. Here, the estimation of the variable is said to depend on the variable of a math condition. For instance, y = 5x + 7. In this equation, the value of variable ‘y’ will change according to the adjustments in the estimation on variable ‘x’. therefore, y variable is said to be a dependent variable 

Independent Variables

According to algebraic expressions, a variable whose values do not change is known as independent changes. They have their own value, which is not changed with the equation. If r and y are the variables in an algebraic expression, the value of x is linked with other values of y, then y is said to be a function of x value known as an independent variable, and they value is known as a dependent variable. 

For instance, in the equation r = b²

Here, r is a dependent, and b is an independent variable. 

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