About Variables in Math
What are Variables?
In algebra, the expressions consist of variables, coefficients, constants, and operators.
A variable is a character that has an unknown value and is represented by an alphabet or a letter. A variable can be any letter from ‘a’ to‘ z’. However, the most commonly used letters used as variables are a, b, x, y, and z .
A coefficient is a numeric value multiplied by a variable.
A constant is a known numeric value that is fixed over time.
An operator is an arithmetic symbol used to perform operations on the values to balance the equation. The most common operators used are plus ( + ), minus ( – ), multiplication ( \(\times\) ), division ( \(\div\) ), equal to ( = ), and so on.
For example, in the equation 2x + 4 = 16, x is the variable that is unknown.
Types of Variables
There are two types of variables:
i) The Dependent Variable:
The variable whose value depends on another variable is called the dependent variable.
ii) The Independent Variable:
The variable whose value does not depend on another variable is called the independent variable.
Example: Consider an equation, y = 5x + 9. Here, the value of variable y will vary with the estimation of the value of variable x. Thus, we can say that y is a dependent variable and x is an independent variable.
Solved Examples on Variables
Example 1: Identify parts of the given algebraic expression: 3a + 5 = 23.
Solution:
Given equation: 3 a + 5 = 23
Here, a → variable
3 → coefficient of a
5 and 23 → constants
+ → operator
Example 2: Evaluate: 3x + 15 = 60.
Solution:
Given equation: 3x + 15 = 60
\(\Rightarrow\) 3x + 15 – 15 = 60 – 15 [Subtract 15 on both sides]
\(\Rightarrow\) 3x = 45
\(\Rightarrow\) x = \(\frac{45}{3}\) [Divide both sides by 3]
x = 15
Therefore, the value of the variable x is 15.
Example 3: Thrice a number less than 36 is equal to 15. Form an algebraic equation with any variable and find the value of the number.
Solution:
Let the number be x
Thrice the number is 3x
Thrice the number less than 36 is 36 – 3x
So, thrice the number less than 36 equal to 15,
36 – 3x = 15
Now solve for the variable x
36 – 3x – 15 = 15 – 15 [Subtract 15 on both sides]
21 – 3x = 0 [Simplify]
3x = 21
x = \(\frac{21}{3}\) [Divide both sides by 3]
x = 7
Therefore, the required expression is 36 – 3x = 15, and the value of the unknown number is 7.
Example 4: Identify the dependent and the independent variables in the given expressions.
i) y = 5x + 6
ii) 2m + 7 = 4n
Solution:
i) In y = 5x + 6, the value of the variable y is dependent on the estimated value of x.
For x = 1, y = 5(1) + 6 = 5 + 6 = 11
For x = 2, y = 5(2) + 6 = 10 + 6 = 16
Therefore, in the given expression, the dependent and independent variables are y and x, respectively.
ii) In 2m + 7 = 4n, the value of the variable n is dependent on the estimated value of m.
For m = 1, n = 2(1) + 7 = 2 + 7 = 9
(2) + 7 = 4 + 6 = 10
Therefore, in the given expression, the dependent and independent variables are n and m, respectively.
In an algebraic expression, both constants and coefficients are numeric values, but a coefficient is multiplied by a variable, whereas a constant is a fixed value.
There are two types of variables: dependent variables and independent variables.
A dependent variable is one that is dependent on the value of another variable in an expression.
Example: x = 3y
Here, the value of x changes with the value of y.
So, y is an independent variable and x is a dependent variable.
An independent variable is one that is not dependent on the value of another variable in an expression.
Example: x = 2y + 1
Here, the value of x is dependent upon the value of y.
So, y is an independent variable and x is a dependent variable.
For y = 0 ⇒ x = 1
For y = 1 ⇒ x = 21 + 1 = 2 + 1 = 3, and so on.
A single mathematical expression is known as a term. It can be a single number or a single variable.
Example: In 2a + 3, there are two terms, that is, 2a and 3, separated by the operator “+.”