Algebraic Expression
An algebraic expression is a mathematical phrase that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. It does not have an equality sign (=) like an equation.
Example:
- 3x + 5 (One variable expression)
- 2a² – 4b + 7(Two variables expression)
- x² + y² – 2xy(Two variables expression)
- 2p − 3q + 4r(Three variables expression)
Here, variables (like x, y, a, b, p, q, r) represent unknown values, and constants (like 3, 5, 7) are fixed numbers. The combination of these elements forms an algebraic expression.
Linear Expressions
A linear expression is an algebraic expression in which the highest power (exponent) of the variable(s) is 1. These expressions do not contain variables with exponents greater than 1 or terms involving the multiplication of variables.
Examples:
- With One Variable:
- 3x + 5
- 7y – 2
- With Two Variables:
- 2x + 3y – 4
- a – 5b + 6
- With Three Variables:
- p + 2q – 3r + 8
Key Features of Linear Expressions:
✔ The exponent of each variable is 1.
✔ There are no terms like x², y³, or xy (no variable multiplication or higher powers).
✔ They do not have an equal sign (=), unlike linear equations.
Equation
An equation is a mathematical statement that shows the equality between two expressions using an equal sign (=). It consists of variables, constants, and mathematical operations, and it can be solved to find the value of the unknown variables.
The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS). In an equation the values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variable. These values are the solutions of the equation.
Examples:
- Linear Equation: 2x + 5 = 9
- Quadratic Equation: x² – 4x + 3 = 0
Linear Equations
A linear equation is an algebraic equation in which the highest power of the variable(s) is 1. The general form of a linear equation in one variable is:
ax + b = 0
where:
- a and b are constants (real numbers),
- x is the variable.
A linear equation in two variables typically has the form:
ax + by = c
where:
- a, b, and c are constants,
- x and y are variables.
Examples:
- One Variable: 2x + 3 = 7
- Two Variables: 3x – 4y = 12
Key Features of Linear Equations:
- The variables have exponents of 1.
- The graph of a linear equation in two variables is a straight line.
- It can have one solution, no solution, or infinitely many solutions depending on the equation.