Some examples of expressions we have so far worked with are | Class 8 Mathematics Notes

2.1 Introduction

In earlier classes, students have encountered algebraic expressions and equations. Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. Examples of such expressions include 5x, 2x - 3, 3x + y, 2xy + 5, xyz + x + y + z, x² + 1, and y + y². Equations, on the other hand, are equalities involving variables and contain an equality sign (=). Examples of equations are 5x = 25, 2x - 3 = 9, 2y + 5/2 = 37/2, and 6z + 10 = -2.

Among these expressions, some have more than one variable, such as 2xy + 5, which has two variables. However, when forming equations, we restrict ourselves to expressions with only one variable. Moreover, the expressions used to form equations are linear, meaning the highest power of the variable appearing in the expression is 1. For example, linear expressions include 2x, 2x + 1, 3y - 7, 12 - 5z, and (5/4)(x - 4) + 10. Expressions like x² + 1, y + y², and 1 + z + z² + z³ are not linear because the highest power of the variable is greater than 1.

This chapter focuses on linear equations in one variable, i.e., equations formed by linear expressions containing only one variable. These are the simple equations studied in earlier classes. An algebraic equation is an equality involving variables, with an equality sign separating the Left Hand Side (LHS) and the Right Hand Side (RHS). The values of the expressions on the LHS and RHS are equal only for certain values of the variable, which are called the solutions of the equation.

For example, x = 5 is a solution of the equation 2x - 3 = 7 because substituting x = 5 gives LHS = 2 × 5 - 3 = 7 = RHS. On the other hand, x = 10 is not a solution because substituting x = 10 gives LHS = 2 × 10 - 3 = 17 ≠ RHS.

To find the solution of an equation, we assume the two sides are balanced and perform the same mathematical operations on both sides so that the balance is not disturbed. A few such steps lead to the solution.

📊 Diagram: Figure 0852062; Figure 0852062; ^{}[] Reprint 2024-25

🧪 Activity: No specific activity in this section; introductory explanation.

🔗 Connection: Leads to the next section which discusses solving equations having variables on both sides.