ALGEBRA: Variable Expressions

James and Natalie were playing with matchsticks and thought of forming patterns of numbers using the matchsticks.

Natalie added 3 more matchsticks to form a pattern with two 4s.

Then James again added 3 more matchsticks to form a pattern with three 4s.

Suddenly, Natalie wondered how many matchsticks are required to make a pattern of ten 4s.

They understood from the existing pattern that they need 4 + 9(3) sticks to get it done as they want a pattern with ten 4s.

From this, they concluded that they need 4 + (n − 1)3 sticks, in general, to make a pattern with n number of 4s.

Here, 4 + (n − 1)3 is called an algebraic expression.

• A symbol that doesn't have a fixed value is called a variable in Math. It can take any value.
  In the above example, n is a variable, and here it can take the values 1, 2, 3, ...
  Some examples of variables in Math are a, b, x, y, z, m, etc.
• A symbol that has a fixed numerical value is called a constant.
  All numbers are constants.
  Some examples of constants are 3, 6, -1/2, 5, etc.
• A term is a variable alone (or) a constant alone (or) it can be a combination of variables and constants by the operation of multiplication or division.
  Some examples of terms are 3x², -2/3y, 5m, etc.
  Here, the numbers that are multiplying the variables are 3, -2/3, and 5, which are called coefficients.

• A variable expression (or) an algebraic expression is a combination of terms by the operations such as addition, subtraction, multiplication, division, etc.
• Example of a Variable Expression:
  An example of a variable expression (or) an algebraic expression is 5X + 7.

To evaluate a variable expression at a given value, we just substitute that value in the expression and simplify it.

Example:
Evaluate 5x² + 2x + 7 at x = -2.

Solution:
Substitute x = -2 in 5x² + 2x + 7:
5(-2)² + 2(-2) + 7 = 5(4) - 4 + 7
= 20 - 4 + 7
= 23
So the answer is 23.

There are 3 main types of algebraic expressions:

• Monomial Expression:
  An algebraic expression that has only one term is known as a monomial.
  Examples of monomial expressions include 3x⁴, 3xy, 3x, 8y, etc.
• Binomial Expression:
  A binomial expression is an algebraic expression that has two terms, which are unlike.
  Examples of binomial expressions include 5xy + 8, xyz + x³, etc.
• Polynomial Expression:
  In general, an expression with more than one term with non-negative integral exponents of a variable is known as a polynomial.
  Examples of polynomial expressions include ax + by + ca, x³ + 2x + 3, etc.

Apart from monomial, binomial, and polynomial types of expressions, an algebraic expression can also be classified into two additional types:

• Numeric Expression:
  A numeric expression consists of numbers and operations but never includes any variable.
  Some examples of numeric expressions are 10 + 5, 15 ÷ 2, etc.
• Variable Expression:
  A variable expression is an expression that contains variables along with numbers and operations to define an expression.
  A few examples of variable expressions include 4x + y, 5ab + 33, etc.

The general algebraic formulas we use to solve expressions or equations are: