Equations in Two Variables

An equation in one variable, like $2x = 6$ , has a single solution. When we introduce a second variable, the situation changes dramatically: a single equation in two variables generally has infinitely many solutions.

Solutions as Ordered Pairs

📐Definition — Solution of an Equation in Two Variables

A solution of an equation in two variables $x$ and $y$ is an ordered pair $(x, y)$ that makes the equation true when substituted. The solution set is the collection of all such ordered pairs.

Consider the equation $2x - y = 4$ . The pair $(3, 2)$ is a solution because $2(3) - 2 = 4$ . But so is $(0, -4)$ , $(2, 0)$ , $(5, 6)$ , and infinitely many others. Each solution is a point in the coordinate plane, and together they form the graph of the equation.

Graphing a Linear Equation

Every equation of the form $ax + by = c$ (where $a$ and $b$ are not both zero) graphs as a straight line. To draw the line, we only need two points. The most convenient choices are usually the intercepts:

x-intercept: set $y = 0$ and solve for $x$
y-intercept: set $x = 0$ and solve for $y$

✎Example — Graphing by Intercepts

Graph $2x - y = 4$ .

Solution. Find the intercepts:

When $x = 0$ : $2(0) - y = 4 \Rightarrow y = -4$ . Point: $(0, -4)$ .
When $y = 0$ : $2x - 0 = 4 \Rightarrow x = 2$ . Point: $(2, 0)$ .

Plot $(0, -4)$ and $(2, 0)$ , then draw the line through them. Every point on this line is a solution to the equation.

Slope-Intercept and Standard Forms

Two common ways to write a linear equation:

Standard form: $ax + by = c$ , where $a, b, c$ are constants.
Slope-intercept form: $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

The slope measures steepness and direction: positive slope means the line rises left to right, negative slope means it falls.

⚡Theorem — Slope Formula

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line with $x_1 \neq x_2$ , the slope is:

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Writing an Equation from Two Points

✎Example — Equation from Two Points

Write the equation of the line through $(1, 3)$ and $(3, 7)$ .

Solution. First, find the slope:

$m = \dfrac{7 - 3}{3 - 1} = \dfrac{4}{2} = 2$

Using point-slope form with the point $(1, 3)$ :

$y - 3 = 2(x - 1) \implies y = 2x + 1$

Special Cases

Not every line fits neatly into slope-intercept form:

Horizontal line $y = k$ : the slope is $0$ . Every point has the same $y$ -coordinate.
Vertical line $x = k$ : the slope is undefined. Every point has the same $x$ -coordinate. This cannot be written as $y = mx + b$ .

✏Exercise — Practice

Graph $3x + 2y = 6$ by finding its intercepts. Then rewrite the equation in slope-intercept form and identify the slope.