Algebra - Variable
- Definition of Variable
- Types of Variables
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Definition of Variable
In real world scenarios, there are things that do not change like your place of birth. The quantities or things that may change are called variables like your grade, score, etc. Any English alphabet or letter can be used to express a variable. However, `x` and `y` are the most commonly used letters to represent a variable.
Given an algebraic expression: `5x + 8`. In the expression, `x` can take any value. If we plug in `x` for `1`, the expression becomes `13`.
Likewise, for `x = 2`, the expression becomes `5(2) + 8 = 18`. We can see that as we change the value of the variable, the value of the expression also changes.
Let’s now consider an equation: `3x + 9 = 15`. Here we can replace `x` with any number. However, the plugged in value of `x` may or may not satisfy the equation. The value that satisfies the equation is called the solution of the equation.
Putting `x = 0`, the equation `3x + 9 = 15` becomes `3(0) + 9 = 15` which is not true. So, `x = 0` is not a solution.
For `x = 1`, the equation `3x + 9 = 15` becomes `3(1) + 9 = 15` which is not true. So, `x = 1` is not a solution.
For `x = 2`, the equation `3x + 9 = 15` becomes `3(2) + 9 = 15` which is true. So, `x = 2` is a solution.
Types of Variables
There are two types of variables:
- Dependent variables: It is a type of variable whose value is determined by the value of another variable. For example, `x` can have any value (`1,2,3,` etc.) in the equation `y = x + 9`. But the value of `y` depends on the value of `x`. Therefore, if `x = 1`, then `y = 10`, when `x = 2, y= 11`. Hence, `y` is called the dependent variable.
- Independent variables: In an algebraic equation, an independent variable is one on which there is no effect of another variable. When there are two variables in an algebraic equation, `x`, and `y`, and the value of `y` depends on the value of `x`. So, `x` is the independent variable, and `y` is the dependent variable. For instance, `x` can have any value in the equation `y = 2x`. Therefore, it is the equation's independent variable.
Solved Examples
Example `1`: Does `x = 6` satisfy the equation `3x - 7 = 14`?
Solution:
When we plug in `x = 6` in the equation, we get `3(6) - 7 =14` which is not a true statement.
So, `x = 6` does not satisfy the equation, meaning `x = 6` is not a solution to `3x - 7 = 14`.
Example `2`: Find the value of the expression `4x + 10` when `x = 2`.
Solution:
`4(2) + 10 = 8 + 10 = 18`
Example `3`: On a quiz practice, students win points for answering questions correctly. The number of points earned is based on how many correct answers they give.
`p =` the number of points earned
`c =` the number of correct answers
Which of the variables is the dependent variable?
Solution:
`p` is the dependent variable as the number of points earned depends on the number of correct answers.
Example `4`: Is `x =12` a solution to the equation `9x - 9 = 99`?
Solution:
When we plug in `x = 12` in the equation, we get `9(12) - 9 = 99` which is a true statement.
So, `x = 12` does satisfy the equation, meaning `x = 12` is a solution to `9x - 9 = 99`.
Practice Problems
Q`1`. What is the value of `y` when `x` is equal to `3` for the equation `y = 8x + 2`.
- `32`
- `24`
- `22`
- `26`
Answer: d
Q`2`. Find out the value of `y` that satisfies the equation `15y = 330`.
- `22`
- `20`
- `21`
- `25`
Answer: a
Q`3`. What is the dependent variable and independent variable in the equation `2a - 9 = b`?
- Dependent variable: \(a\), Independent variable: \(b\)
- Dependent variable: \(b\), Independent variable: \(a\)
- Both \(a\) and \(b\) are dependent variables
- Both \(a\) and \(b\) are independent variables
Answer: b
Q`4`. Maria is bringing candies to her gymnastics class on her birthday. She decides the number of candies she needs to buy by counting the number of students in her class.
`s =` the number of students
`c =` the number of candies
Which of the variables is the independent variable?
- `s`
- `c`
- both `s` and `c`
- None of these
Answer: a
Frequently Asked Questions
Q`1`. What is a variable in mathematics?
Answer: A variable in mathematics is a symbol or letter used to represent an unknown or changing quantity in an equation or expression. It can take on different values, and its purpose is to generalize mathematical relationships.
Q`2`. What is the difference between independent and dependent variables?
Answer: An independent variable is a variable that is manipulated or changed in an experiment or equation, while a dependent variable is a variable that responds to the changes in the independent variable.
Q`3`. How are variables used in algebraic expressions?
Answer: In algebraic expressions, variables are used to represent unknown values or quantities. They allow for the generalization of mathematical relationships and are essential for solving equations and inequalities.