 ## What are variables and expressions

So, let’s understand the meaning of a variable first. A variable can be expressed as an entity which can have any value from negative infinity to positive infinity. Suppose there is a variable $$n$$ which is a set of natural numbers. So, the value of $$n$$ can be a subset of integers $$(1, \ 2, \ 3, \ ……….)$$. As you can see that $$n$$ can take any values ranging from $$1$$ to infinity. So, $$n$$ is a variable. In mathematics, you will see the use of variables in any alphabet like $$a, \ b, \ x, \ y, \ z, \ m$$, etc.
Now, since you have understood the term variable, let’s learn about a constant. As the name suggests, a constant is a term that has a fixed value. For example, $$5, \ 6, \ 0.3, \ \frac{3}{5}$$ are all constants.
Now, in mathematics, a term can be either completely constant or completely a variable, or a mixture of the two. For example, let’s take the term $$2z$$. Here $$z$$ is the variable and of course $$2$$ is the constant term. Also, $$2$$ is called the co-efficient of $$z$$. Some other examples of such terms are $$-3y, \ 8x, \ \frac{2}{9} \ a$$, etc.
Variables, constants, and their mixtures are all identified as terms in mathematics.

### Expression

A variable expression or an algebraic expression is the combination of terms (constants, variables, or both) and is separated by arithmetic operations like addition, subtraction, multiplication, or division.
Some examples of expressions are: $$5x \ + \ 7, \ 2x^2 \ + \ 7x \ – \ 6, \frac{6}{7} \ x \ - \ 4$$, etc.
Let’s see the use of some variables in mathematical operations:

• Addition: $$4 \ + \ x$$ ($$4$$ plus $$x$$)
• Subtraction: $$2 \ – \ a$$ ($$2$$ minus $$a$$)
• Division: $$4 \div x$$ ($$4$$ divided by $$x$$)
• Multiplication: $$7 \times b$$ ($$7$$ multiplied by $$b$$)

Ex: Solve for $$x = 3$$, $$x^3 \ – \ 2x^2 \ + \ 9x \ + \ 7$$
So, here we just put $$x \ = \ 3$$ to get, $$3^3 \ – \ 2(3)^2 \ + \ 9(2) \ + \ 7 \ = \ 27 \ – \ 18 \ + \ 18 \ + \ 7 \ = \ 34$$.

### Exercises for Expressions and Variables

1) Find $$10 \ + \ x(21x \ - \ 1)$$, if $$x \ = \ 1 \ \Rightarrow \$$

2) Find $$10 \ + \ x(14x \ - \ 4)$$, if $$\ x \ = \ 4 \ \Rightarrow \$$

3) Find $$x \ + \ 16x$$, if $$\ x \ = \ 5 \ \Rightarrow \$$

4) Find $$x \ + \ 2x$$, if $$\ x \ = \ 3 \ \Rightarrow \$$

5) Find $$x \ + \ 9x$$, if $$\ x \ = \ 8 \ \Rightarrow \$$

6) Find $$x \ + \ 11x \ - \ 6$$, if $$\ x \ = \ 6 \ \Rightarrow \$$

7) Find $$x \ + \ 18x \ - \ 1$$, if $$\ x \ = \ 1 \ \Rightarrow \$$

8) Find $$x \ + \ x(13x \ - \ 1)$$, if $$\ x \ = \ 1 \ \Rightarrow \$$

9) Find $$x \ + \ x(20x \ - \ 4)$$, if $$\ x \ = \ 4 \ \Rightarrow \$$

10) Find $$x \ + \ x(21x \ - \ 1)$$, if $$\ x \ = \ 1 \ \Rightarrow \$$

1) Find $$10 \ + \ x(21x \ - \ 1)$$, if $$x \ = \ 1 \ \Rightarrow \$$$$\color{red}{10 + 1(21 \ \times 1 \ - \ 1) \ = 30}$$
2) Find $$10 \ + \ x(14x \ - \ 4)$$, if $$\ x \ = \ 4 \ \Rightarrow \$$$$\color{red}{10 + 4(14 \ \times 4 \ - \ 4) \ = 218}$$
3) Find $$x \ + \ 16x$$, if $$\ x \ = \ 5 \ \Rightarrow \$$$$\color{red}{5 + 16 \ \times 5 \ = 85}$$
4) Find $$x \ + \ 2x$$, if $$\ x \ = \ 3 \ \Rightarrow \$$$$\color{red}{3 + 2 \ \times 3 \ = 9}$$
5) Find $$x \ + \ 9x$$, if $$\ x \ = \ 8 \ \Rightarrow \$$$$\color{red}{8 + 9 \ \times 8 \ = 80}$$
6) Find $$x \ + \ 11x \ - \ 6$$, if $$\ x \ = \ 6 \ \Rightarrow \$$$$\color{red}{6 + 11 \ \times 6 \ - \ 6 \ = 66}$$
7) Find $$x \ + \ 18x \ - \ 1$$, if $$\ x \ = \ 1 \ \Rightarrow \$$$$\color{red}{1 + 18 \ \times 1 \ - \ 1 \ = 18}$$
8) Find $$x \ + \ x(13x \ - \ 1)$$, if $$\ x \ = \ 1 \ \Rightarrow \$$$$\color{red}{1 + 1(13 \ \times 1 \ - \ 1) \ = 13}$$
9) Find $$x \ + \ x(20x \ - \ 4)$$, if $$\ x \ = \ 4 \ \Rightarrow \$$$$\color{red}{4 + 4(20 \ \times 4 \ - \ 4) \ = 308}$$
10) Find $$x \ + \ x(21x \ - \ 1)$$, if $$\ x \ = \ 1 \ \Rightarrow \$$$$\color{red}{1 + 1(21 \ \times 1 \ - \ 1) \ = 21}$$

## Expressions and Variables Practice Quiz

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