Introduction to Linear Equations

What are Linear Equations?

An equation is a mathematical statement that says two expressions are equal. When the highest power (degree) of the variable in an equation is exactly 1, it is called a linear equation. The word "linear" comes from the fact that these equations represent straight lines when plotted on a graph. Linear equations are one of the most important topics in Class 8 Mathematics, tested across CBSE, Telangana, and Andhra Pradesh board exams.

• 2x + 3 = 5 — linear (degree of x is 1)
• a + 2b + 9 = 0 — linear (all variables have degree 1)
• l − 3m = 5n — linear (all variables have degree 1)
• 5x² + 6xy − 4y² = 16 — NOT linear (degree is 2)
• xy + yz + zx = 11 — NOT linear (products of variables give degree 2)

Linear Equation in One Variable

When a linear equation contains only one variable, it is called a linear equation in one variable, also known as a simple equation. This is the main focus of Chapter 2. The general form is ax + b = c, where a, b, and c are numbers and x is the single unknown variable.

• 3m + 7 = 13 — simple equation (one variable: m)
• 7y − 4 = 2y + 1 — simple equation (one variable: y)
• x/3 − x/5 = 2 — simple equation (one variable: x)
• 3 = 2x + y — NOT a simple equation (two variables: x and y)
• x² + 5x + 3 = 0 — NOT a simple equation (degree is 2)
• 5m − 6n = 0 — NOT a simple equation (two variables: m and n)

How to Check Whether an Equation is Linear in One Variable

Use this two-step test: first, check that the highest power of any variable is 1 (no squares, cubes, or products of variables). Second, check that only one variable appears in the equation. If both conditions are met, it is a linear equation in one variable.

Solution of an Equation

The solution (or root) of an equation is the value of the variable that makes the Left Hand Side (LHS) equal to the Right Hand Side (RHS). Finding this value is the core skill of this chapter.

Consider 3x − 5 = 7. Testing x = 5: LHS = 3(5) − 5 = 10, but RHS = 7. Since 10 ≠ 7, x = 5 is not a solution. Testing x = 4: LHS = 3(4) − 5 = 7, and RHS = 7. Since LHS = RHS, x = 4 is the solution.

Key Terms to Remember

• Degree of an equation — the highest power of the variable. Linear equations have degree 1.
• Variable — the unknown quantity, usually represented by letters like x, y, or m.
• LHS and RHS — the expressions to the left and right of the equals sign.
• Root / Solution — the value of the variable that satisfies the equation (makes LHS = RHS).
• Simple equation — another name for a linear equation in one variable.

What This Lesson Prepares You For

Understanding what a linear equation in one variable is and how to verify a solution forms the foundation for everything else in Chapter 2. The next steps involve learning systematic methods to solve these equations — transposing terms, cross-multiplication, and handling equations with fractions. These skills are built up through the exercises and are directly useful in Exercise 2.1 and beyond. The same concepts extend into polynomials and quadratic equations in Class 9 and 10. For related foundational work, see the Introduction to Rational Numbers, as solving equations often involves rational number operations.