Chapter 4: Linear Equations in Two Variables

In earlier classes you solved equations like $2x + 5 = 11$ and got a single answer, $x = 3$ . With two variables, the story changes dramatically. The equation $2x + 3y = 6$ has not one, not two, but infinitely many solutions , and they all line up on a straight line in the Cartesian plane. This chapter is your introduction to that idea.

The chapter is short but conceptually rich. You will learn the standard form $ax + by + c = 0$ , understand what counts as a solution, generate tables of solutions, and most importantly graph the equation as a line. The connection between an equation and its graph is the headline takeaway: the equation is the algebra; the line is the geometry. Same object, two languages.

You will also examine a few special cases that often confuse students. Equations like $x = 4$ or $y = -2$ also "live in two variables" once you allow them on the coordinate plane , they are vertical and horizontal lines respectively. We sort these out clearly: lines parallel to the $y$ -axis have the form $x = c$ , and lines parallel to the $x$ -axis have the form $y = c$ . The general form $ax + by + c = 0$ subsumes both.

Why care? Linear equations describe a huge slice of real-world relationships: distance covered at constant speed, salary plus a fixed bonus, the conversion between Celsius and Fahrenheit, the budget line of a shopper deciding how to split money between two items. Recognising a problem as "linear in two variables" lets you solve it with a graph in two minutes. In Class X you will see pairs of such equations and how their graphs intersect; in Class XI you will see lines parameterised differently. The foundation is laid here.

A final note: in this chapter we draw a single line per equation. Next year you will draw pairs of lines and find where they cross. So pay attention to graphing technique now , it is the skill on which the next chapter rests.

What's inside

Linear equations in two variables , the standard form and what counts as one.
Solutions of a linear equation , what an ordered pair $(x, y)$ must satisfy, and the infinitude of solutions.
Graphing a linear equation , turning the solution set into a straight line.
Lines parallel to the axes , special cases $x = c$ and $y = c$ .
Linear models of real situations , translating word problems into equations and graphs.

Key results / Formula card

Concept	Statement
Standard form	$ax + by + c = 0$ , where $a, b, c$ are real and $a, b$ are not both zero.
Solution	An ordered pair $(p, q)$ is a solution iff $ap + bq + c = 0$ .
Number of solutions	A linear equation in two variables has infinitely many solutions.
Graph	The graph of $ax + by + c = 0$ in the plane is a straight line.
Line parallel to $y$ -axis	$x = c$ (for some constant $c$ ). All such points share the same abscissa.
Line parallel to $x$ -axis	$y = c$ (for some constant $c$ ). All such points share the same ordinate.
$x$ -intercept	The point where the line meets the $x$ -axis, found by setting $y = 0$ .
$y$ -intercept	The point where the line meets the $y$ -axis, found by setting $x = 0$ .
Two points determine a line	To graph $ax + by + c = 0$ , plot any two solutions and join them.

Keep this card next to you while working through the chapter. Every problem reduces to one of these statements.

Chapter 4: Linear Equations in Two Variables

What's inside

Key results / Formula card

Sub-topics

Practice quiz