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,

c, 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: \(a(b + c) = ab + ac\)

\(5x(3x + 4) = (5x × 3x) + (5x × 4)\\[1em] 5x(3x + 4)= 15x^2 + 20x\)

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, \(24x^2 + 36x\) by \(6x\)

\((\frac{24x^2}{6x}) + (\frac{36x}{6x}) = 4x + 6\)

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

If \(x = 2\):

\(5x^2 + 5x = 5(2)^2 + 5(2) = 5(4) + 10 = 20 + 10 = 30\)

If \(x = 5\): 

\(5x^2 + 5x = 5(5)^2 + 5(5) = 5(25) + 25 = 125 + 25 = 150\)

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)

\(18(2x + 3) = (18 × 2x) + (18 × 3)\\[1em] 18(2x + 3)= 36x + 54\)

Variable expressions are algebraic phrases that include letters (variables) and numbers—means learning to manipulate, simplify, and understand how variables behave under different operations. Here are the best tips and tricks to help you master them.
 

1. Make sure to combine like terms before solving the equations. Like terms have the same variable raised to the same power. Circle or color-code like terms when simplifying long expressions.
    
2. Use the distributive property, as the distributive law lets you remove parentheses. \(a (b + c) = ab + ac\)

   
   Example:\(3 (x + 4) = 3x + 12\)

   
   Reverse it to factor:\(6x+12=6(x+2)\)
    

3. Keep the “Invisible 1” in your mind. Every variable has an invisible 1 (𝑥 = 1𝑥). This helps when subtracting or dividing variables
    

4. Think of expressions as machines. An expression is like a machine that takes an input (the variable) and gives an output. This mindset helps while substituting values, simplifying functions, graphing relationships.
    

5. Check units or context in word problems. If 𝑥 represents apples and 𝑦 represents dollars, then \(3x+5y\) makes no sense (you can’t add apples and dollars). Always match what each variable means.

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. 
 

1. 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. 
    
2. 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.
    
3. Home Expenses: If your monthly rent is $800, and utilities cost $50 per person for 𝑛 roommates.
   
   Expression: \(T=800+5n\)
    
4. 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 \(10x + 500\).
    
5. Travel and distance: A car travels at 60 km/h for t hours. Its expression is given as \(d=60t\). The distance depends on the time driven. When there's a 10km head star, \(d = 60t + 10\)