Variable Expressions

In algebraic expressions, variables are the symbols, mostly English letters like x, y, and z. They represent the unknown or changing value; that is, it depends on the expression. For example, 5x + 5, where x is the variable.  
 

Expressions with variables are mathematical expressions that combine one or more variables, numbers, and operations. Variables are symbols used to represent unknown or changeable values. For example, in the expression 5x + 3y -10, x and y are the variables.  
 

Algebraic expressions can be classified into two categories based on the number of variables they have. The types are: 

• Expressions with one variable
• Expressions with multiple variables

Expressions with One Variable 
The expressions that contain only one variable are called expressions with one variable. For example,
25x + 15, where x is the variable, 25 is the coefficient, and 15 is the constant
2y + 6, where y is the variable, 2 is the coefficient, and 6 is the constant  
The value of these expressions changes as the value of the variable changes. 

If x = 5, the value of 25x + 15 = 25(5) + 15 = 125 + 15 = 140
If x = 6, the value of 25x + 15 = 25(6) + 15 = 150 + 15 = 165

Expressions with Multiple Variables
Expressions with multiple variables are expressions that have more than one variable. Here, the solution of the expression changes based on the values of the variables. For example, 2x + 4y - 8, where x and y are the variables, 2 and 4 are the coefficients, and -8 is the constant. 

If x = 2 and y = 1, 2x + 4y - 8 becomes: 2(2) + 4(1) - 8 = 4 + 4 - 8 = 0
If x = 6 and y = 2, 2x + 4y - 8 becomes: 2(6) + 4(2) - 8 = 12 + 8 - 8 = 12
 

Operations on expressions involve applying mathematical procedures such as addition, subtraction, multiplication, division, and factoring to expressions involving one or more variables. The main types of operations are: 

• Adding and subtracting expressions
• Multiplication of expressions
• Division of expressions
• Solving expressions
• Factoring expressions
• Expanding expressions

Adding and Subtracting Expressions: Addition and subtraction are the basic arithmetic operations. In addition, we combine the like terms in the expressions. 
For example, (3x + 5) + (5x + 4) = (3x + 5x) + (5 + 4) = 8x + 9

In subtracting expressions, the like terms from the second expression are subtracted from the corresponding like terms from the first expression. 
For example, (6x+ 3) - (2x + 1) = (6x - 2x) + (3 - 1) = 4x + 2

Multiplication of Expressions
To multiply expressions, we use the distributive property of multiplication, that is, a(b + c) = ab + ac. 
For example, multiplying 3x + 4 with 5x
Using the distributive property of multiplication: a(b + c) = ab + ac
5x(3x + 4) = (5x × 3x) + (5x × 4)
= 15x2 + 20x

Division of Expressions: The division of expressions is used to simplify the expressions. Here, the expressions are divided by separate terms, by factoring or simplifying common terms. 
For example, 24x2 + 36x by 6x
(24x2/6x) + (36x/6x) = 4x + 6

Solving Expressions: Solving expressions involves substituting the variables with the given value to find the result. For example, solve 5x2 + 5x for x = 2 and x = 5

If x = 2:
5x2 + 5x = 5(2)2 + 5(2) = 5(4) + 10 = 20 + 10 = 30
If x = 5: 
5x2 + 5x = 5(5)2 + 5(5) = 5(25) + 25 = 125 + 25 = 150

Factoring Expressions: Factoring expressions is the way of expressing an expression as the product of its factors. In this method, we first factor out the greatest common factor of the expression. 

For example, factoring 36x + 54
The GCF of 36 and 54 is 18
So, 36x + 54 = 18(2x + 3)

Expanding Expressions: Expanding expressions is the opposite of factoring, as here we remove the parentheses by multiplying the value out of the expression with the expression inside the parentheses. 
For example, expanding the expression 18(2x + 3)
18(2x + 3) = (18 × 2x) + (18 × 3)
= 36x + 54
 

Variable expression is used in different fields like mathematics, physics, finance, planning, etc., to find the value of unknown or changing values. In this section, we will explore some real-life applications of variable expressions. 

• In budgeting, variable expressions are used to calculate the cost, savings, or any quantities that can change. For example, when shopping, to calculate the cost of buying multiple items can be calculated using the variable expressions. 
• In cooking, we use variable expressions to adjust the recipe based on the number of people we are serving. For example, if we need 0.5x cups of flour, where x is the cups per person. If we need to cook for 4 people, then x = 4: 
• 0.5x = 0.5(4) = 2 cups of flour.
• Variable expression is used in travel planning to calculate the time required to travel by adjusting the speed and distance. 
• In business, to calculate the profit, production cost, and revenue, we use a variable expression. For example, if the production cost for x items is $10 and the fixed cost is $500, then the total cost is 10x + 500