Graphing Linear Equations

If you have ever looked at an equation and wondered, “But what does this actually mean?” then graphing is your answer. Graphing is the bridge between abstract symbols and something you can actually see. When you graph a linear equation, you are not just drawing a line - you are revealing everything that equation has to say. Every point on that line is a solution to the equation, and the line itself tells a story about how two quantities relate to each other.

Think of it this way: an equation like $y = 2x + 3$ is a rule. It says, “Take any number, double it, and add 3.” But when you graph it, you get to see all the results of that rule at once, stretched out as a beautiful straight line across the coordinate plane.

Core Concepts

Coordinate Plane Review

Before we dive into graphing, let us make sure you are comfortable with the coordinate plane. The coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They meet at the origin, $(0, 0)$.

Every point on the plane has an address written as an ordered pair $(x, y)$:

• The x-coordinate tells you how far to go left or right from the origin
• The y-coordinate tells you how far to go up or down

When we graph an equation, we are finding all the points $(x, y)$ that make the equation true and plotting them. For linear equations, those points always form a straight line - that is why they are called “linear.”

Graphing from a Table of Values

The most straightforward way to graph any equation is to make a table of values. You pick several values for $x$, calculate the corresponding $y$ values, plot those points, and connect them with a line.

For example, to graph $y = x + 2$:

$x$	$y = x + 2$	Point
$-2$	$-2 + 2 = 0$	$(-2, 0)$
$0$	$0 + 2 = 2$	$(0, 2)$
$2$	$2 + 2 = 4$	$(2, 4)$

Plot these three points and draw a straight line through them. Two points are actually enough to define a line, but using three gives you a safety check - if they do not line up, you made a calculation error somewhere.

Slope-Intercept Form: $y = mx + b$

This is the most useful form for graphing because it tells you exactly what the line looks like:

• $m$ is the slope - how steep the line is, and whether it goes up or down
• $b$ is the y-intercept - where the line crosses the y-axis

In $y = 2x + 3$:

• The slope $m = 2$ means the line rises 2 units for every 1 unit you move right
• The y-intercept $b = 3$ means the line crosses the y-axis at the point $(0, 3)$

Graphing Using Slope and Y-Intercept

This method is fast and elegant:

Step 1: Find the y-intercept $b$ and plot the point $(0, b)$.

Step 2: Use the slope $m$ to find another point. Remember, slope is “rise over run”: $$m = \frac{\text{rise}}{\text{run}}$$

If $m = 2$, think of it as $\frac{2}{1}$: rise 2, run 1. From your y-intercept, go up 2 and right 1 to find the next point.

If $m = -\frac{3}{4}$: rise $-3$ (go down 3), run 4 (go right 4).

Step 3: Draw a straight line through both points.

X-Intercept and Y-Intercept

Every line (except horizontal and vertical ones) crosses both axes:

• The y-intercept is where the line crosses the y-axis. At this point, $x = 0$. To find it, substitute $x = 0$ into the equation.

• The x-intercept is where the line crosses the x-axis. At this point, $y = 0$. To find it, substitute $y = 0$ into the equation and solve for $x$.

These intercepts are incredibly useful. They tell you the starting point (y-intercept) and the “break-even” or “zero” point (x-intercept) in many real-world situations.

Standard Form: $Ax + By = C$

Not all linear equations come neatly packaged as $y = mx + b$. Standard form looks like $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is usually positive.

For example: $3x + 2y = 12$

You can always convert standard form to slope-intercept form by solving for $y$, but sometimes it is easier to graph standard form directly using intercepts.

Graphing Using Intercepts

This is often the fastest way to graph equations in standard form:

Step 1: Find the y-intercept by setting $x = 0$ and solving for $y$.

Step 2: Find the x-intercept by setting $y = 0$ and solving for $x$.

Step 3: Plot both intercepts and draw a line through them.

For $3x + 2y = 12$:

• Y-intercept: $3(0) + 2y = 12 \rightarrow y = 6$, so the point is $(0, 6)$
• X-intercept: $3x + 2(0) = 12 \rightarrow x = 4$, so the point is $(4, 0)$

Plot $(0, 6)$ and $(4, 0)$, connect them, and you have your line.

Horizontal and Vertical Lines

These are special cases that often confuse students, but they are actually the simplest lines to graph:

Horizontal lines have the equation $y = k$ where $k$ is some constant. For example, $y = 4$ is a horizontal line passing through all points where the y-coordinate is 4. The slope of a horizontal line is 0 - the line does not rise or fall at all.

Vertical lines have the equation $x = k$. For example, $x = -2$ is a vertical line passing through all points where the x-coordinate is $-2$. The slope of a vertical line is undefined because you would be dividing by zero (the run is 0).

Here is an easy way to remember: $y = 4$ is horizontal because it is saying “y is always 4, no matter what x is.” The line hugs the y-value of 4. Similarly, $x = -2$ is vertical because x is stuck at $-2$ while y can be anything.

Notation and Terminology

Term	Meaning	Example
Slope-intercept form	$y = mx + b$	$y = 2x + 3$
Standard form	$Ax + By = C$	$2x + 3y = 6$
Y-intercept	Where line crosses y-axis (when $x=0$)	In $y = 2x + 3$, $b = 3$
X-intercept	Where line crosses x-axis (when $y=0$)	In $y = 2x - 6$, x-intercept is 3
Horizontal line	Slope = 0	$y = 4$
Vertical line	Undefined slope	$x = -2$

Examples

Key Properties and Rules

Converting Between Forms

From Standard Form to Slope-Intercept Form:

Starting with $Ax + By = C$, solve for $y$: $$By = -Ax + C$$ $$y = -\frac{A}{B}x + \frac{C}{B}$$

So the slope is $m = -\frac{A}{B}$ and the y-intercept is $b = \frac{C}{B}$.

From Slope-Intercept Form to Standard Form:

Starting with $y = mx + b$, rearrange: $$-mx + y = b$$ $$mx - y = -b$$

Multiply through by any constant to get integer coefficients if needed.

Quick Reference for Graphing Methods

Method	Best Used When	Steps
Table of values	Learning the basics; checking work	Pick x-values, calculate y-values, plot points
Slope and y-intercept	Equation is in $y = mx + b$ form	Plot $(0, b)$, use slope to find second point
Intercepts	Equation is in standard form	Find $(0, y)$ and $(x, 0)$, connect them

Slope Quick Guide

Slope	Line Direction	Example
Positive $(m > 0)$	Goes upward left to right	$y = 2x + 1$
Negative $(m < 0)$	Goes downward left to right	$y = -3x + 5$
Zero $(m = 0)$	Horizontal	$y = 4$
Undefined	Vertical	$x = -2$

Real-World Applications

Cost Functions

Many business situations involve fixed costs plus variable costs. A phone plan might charge $30 per month (fixed) plus $0.10 per text (variable). This can be modeled as: $$C = 0.10t + 30$$

The y-intercept (30) represents the base monthly cost. The slope (0.10) represents the cost per text. Graphing this shows you exactly how your bill grows with usage.

Distance-Time Graphs

When something moves at a constant speed, its distance over time forms a linear graph. If you drive at 60 mph: $$d = 60t$$

The slope is the speed (60 mph), and the y-intercept is 0 (you start at distance zero). Reading the graph tells you how far you have traveled at any point in time.

Temperature Conversions

The formula to convert Celsius to Fahrenheit is: $$F = \frac{9}{5}C + 32$$

This is a linear equation! The y-intercept (32) tells you that $0°C = 32°F$. The slope ($\frac{9}{5}$) tells you that Fahrenheit increases by 9 degrees for every 5-degree increase in Celsius.

Depreciation

When you buy a car for $25,000 and it loses $3,000 in value each year, the value over time is: $$V = -3000t + 25000$$

The negative slope shows the value decreasing. The y-intercept is the original purchase price. The x-intercept tells you when the car’s value reaches zero (in this case, after about 8.3 years).

Self-Test Problems

Summary

• Graphing transforms equations into visual representations where you can see all solutions at once.
• A table of values is the most basic method: pick x-values, calculate y-values, plot points, connect with a line.
• Slope-intercept form $y = mx + b$ is ideal for graphing: $b$ is where to start on the y-axis, and $m$ tells you how to move to the next point.
• Slope is rise over run. Positive slopes go up, negative slopes go down, zero slope is horizontal, undefined slope is vertical.
• The y-intercept is where the line crosses the y-axis (when $x = 0$).
• The x-intercept is where the line crosses the x-axis (when $y = 0$).
• Standard form $Ax + By = C$ is easy to graph using the intercept method: find both intercepts and connect them.
• Horizontal lines have the form $y = k$ (slope = 0).
• Vertical lines have the form $x = k$ (slope is undefined).
• Linear equations model many real-world situations: costs, distances, conversions, and depreciation all create straight-line graphs.