2.1. Introduction

In the earlier classes, you have come across several algebraic expressions and equations. Some examples of expressions we have so far worked with are: 5x, 2x – 3, 3x + y, 2xy + 5, xyz + x + y + z, x2 + 1, y + y2. Some examples of equations are: 5x = 25, 2x – 3 = 9, 2y + = ,6z +10= −2. You would remember that equations use the equality (=) sign; it is missing in expressions.

This chunk introduces students to the fundamental concepts of algebraic expressions and equations. An algebraic expression is a combination of numbers, variables, and operations. Examples given, like '5x' and '2xy + 5', show different forms of expressions. In contrast, equations have an equality sign (like '5x = 25'), indicating that both sides represent the same value. Students should note that while expressions do not use the equality sign, equations do, establishing a balance between two quantities.

Consider the difference between a recipe (expression) and a cooking instruction (equation). The recipe tells you what ingredients you need but doesn’t say how much you’ll end up with — that’s like an expression. The cooking instruction, which instructs you to combine ingredients in a specific way to achieve a final dish — indicating a relationship between inputs and outputs — resembles an equation.

Of these given expressions, many have more than one variable. For example, 2xy + 5 has two variables. We however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.

This chunk explains the concept of linear expressions, which are characterized by having only one variable raised to the first power. For example, '2x' and '3y – 7' are linear, while 'x² + 1' is not because it has a variable raised to the second power. In the context of equations, using only one variable simplifies finding solutions and analyzing relationships.

Think of a straight road that represents a linear equation. If you are walking straight, your distance increases consistently, without any curves (no higher powers). If you took a curvy path (like a quadratic), your distance could change unpredictably. Linear expressions ensure predictable outcomes, just like walking in a straight line.

These are linear expressions: 2x, 2x + 1, 3y – 7, 12 – 5z, (x – 4)+10. These are not linear expressions: x2 + 1, y + y2, 1 + z + z2 + z3 (since highest power of variable > 1). Here we will deal with equations with linear expressions in one variable only. Such equations are known as linear equations in one variable.

In this chunk, the text emphasizes differentiating between linear and non-linear expressions. The examples help clarify what makes an expression linear. A linear equation involves expressions where the highest exponent of the variable is 1. By limiting focus to one variable, the study of these equations becomes manageable and directly applicable to real-world situations.

Imagine you're budgeting for a party. If you say, 'I will spend $10 per person,' that's like a linear equation – it gives you a predictable cost based on the number of people. If you say, 'My budget will increase in a non-linear way based on how many people I invite' without a clear rule, that's more complex, like a non-linear expression.

Let us briefly revise what we know: (a) An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression 2x – 3 = 7 on the right of the equality sign is the Right Hand Side (RHS). (b) In an equation the values of the expressions on the LHS and RHS are equal.

Here, students are reminded of the fundamental components of equations. The LHS and RHS terms must balance, meaning whatever adjustments made to one side should reflect on the other to maintain equality. By emphasizing terms like LHS and RHS, students gain an understanding of how to approach solving equations step-by-step.

Think of a balanced scale where two sides must match for it to be level. If you add a weight (like an operation) to one side, you have to add the same weight to the other side to keep it balanced, similar to how equations require maintaining equality between LHS and RHS.

How to find the solution of an equation? We assume that the two sides of the equation are balanced. We perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. A few such steps give the solution.

This chunk provides guidance on how to approach solving an equation. Since both sides are assumed to be equal, operations (like adding or subtracting) applied to one side must also be applied to the other to maintain equality. Eventually, through a series of steps, students can isolate the variable and find its value.

Imagine balancing your checkbook. If you add a transaction on one side to see how much money you have left, you must record that same transaction on the other side. This ensures your balance remains accurate, just like performing equal operations in an equation to solve for a variable.

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