4.1. Introduction

In earlier classes, you have studied linear equations in one variable. Can you write down a linear equation in one variable? You may say that x + 1 = 0, x + 2 = 0 and 2y + 3 = 0 are examples of linear equations in one variable. You also know that such equations have a unique (i.e., one and only one) solution. You may also remember how to represent the solution on a number line.

In previous studies, we learned about linear equations involving a single variable, like the examples given: x + 1 = 0 or 2y + 3 = 0. A linear equation in one variable means that there is only one unknown (either x or y) that can be solved to find a specific value. For example, if we take x + 1 = 0, we can rearrange it to find that x = -1. This unique solution can be plotted on a number line, where each value corresponds to a position.

Think of a simple example: if you have a balance scale and you know one side weighs exactly 0 grams, if you add 1 gram on one side, the other side must also have exactly 1 gram to balance. This is similar to finding the unique solution in a linear equation.

In this chapter, the knowledge of linear equations in one variable shall be recalled and extended to that of two variables. You will be considering questions like: Does a linear equation in two variables have a solution? If yes, is it unique? What does the solution look like on the Cartesian plane?

This chapter builds on what we already know about linear equations. We will explore linear equations that involve two variables, such as x and y, and investigate key questions. One important aspect is whether these equations yield one solution, no solutions, or infinitely many answers. Additionally, we will visualize how the solutions of these equations can be represented on a Cartesian plane, which has both an x-axis and a y-axis.

Imagine plotting the routes of two different cars on a map. Each car's destination could be expressed as a linear equation. Depending on where the cars start and finish, the routes might intersect (indicating a solution) or they might not meet at all (indicating no solution). By studying these equations, we can determine where and if they cross.

You shall also use the concepts you studied in Chapter 3 to answer these questions.

As we delve into linear equations in two variables, we will revisit some concepts from the previous chapter (Chapter 3). This helps reinforce our understanding, as the techniques and ideas from prior learnings will be necessary for solving the equations in this chapter. Familiarity with these concepts gives us the tools we need to tackle more complex equations involving two variables.

Consider learning a recipe: if you first learn how to bake bread (Chapter 3), those skills (like measuring ingredients and kneading dough) will be important when you later try to make a cake that requires similar techniques. Similarly, our previous learnings will be crucial when we work with two-variable equations.

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