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Graphing Linear Equations: Slope, Intercepts, and Reading a Line

Learn how to graph linear equations using slope-intercept form, find slope and intercepts, and read a line's equation directly from its graph.

Practice Graphing

Slope-intercept form

Every linear equation has a graph — and that graph is always a straight line. Once you can read an equation and picture its line (or go the other way, from a line to its equation), a huge amount of Algebra 2 and later math gets easier, because slope and intercepts show up constantly in real-world modeling problems.

The easiest form to graph from is:

y = mx + b

where m is the slope (how steep the line is, and which direction it goes) and b is the y-intercept (where the line crosses the y-axis, at the point (0, b)).

What slope actually means

Slope is "rise over run" — how much y changes for every 1-unit change in x. A slope of m = 2 means the line goes up 2 units for every 1 unit it moves right. A negative slope means the line goes downward left to right. A slope of 0 is a flat, horizontal line.

Worked Examples

Example 1: Graphing from slope-intercept form

Easy

Graph: y = 2x + 3

Step 1: Identify slope and y-intercept
m = 2, b = 3
In y = 2x + 3, the number multiplied by x is the slope (m), and the constant is the y-intercept (b).
Step 2: Plot the y-intercept
Point (0, 3)
Start by plotting the y-intercept on the y-axis.
Step 3: Use slope to find a second point
Up 2, Right 1 -> Point (1, 5)
The slope is 2 (or 2/1). From (0, 3), rise 2 and run 1 to reach (1, 5). Draw a line through both points.
Final Answer:
Line through (0, 3) and (1, 5)

Example 2: Finding slope from two points

Medium

Find the slope of the line through (1, 2) and (4, 8).

Step 1: Use the slope formula
m = (y₂ - y₁) / (x₂ - x₁)
Slope is the change in y over the change in x (rise over run).
Step 2: Substitute the points
m = (8 - 2) / (4 - 1)
Plug in the coordinates (1, 2) and (4, 8).
Step 3: Simplify
m = 6 / 3 = 2
The slope is 2.
Final Answer:
m = 2

Example 3: Going from a graph back to an equation

Hard

A line crosses the y-axis at (0, -1) and passes through (2, 3).

Step 1: Find the slope
m = (3 - (-1)) / (2 - 0) = 4 / 2 = 2
Use the slope formula with points (0, -1) and (2, 3).
Step 2: Identify the y-intercept
b = -1
The line crosses the y-axis at (0, -1), so b is -1.
Step 3: Write the equation
y = 2x - 1
Substitute m = 2 and b = -1 into y = mx + b.
Final Answer:
y = 2x - 1

Common mistakes to avoid

  • Flipping rise and run. Slope is vertical change over horizontal change — rise/run, never the reverse.
  • Misreading the y-intercept. The y-intercept is where the line crosses the y-axis (x = 0), not the x-axis.
  • Forgetting the sign of the slope. A line that goes down from left to right always has a negative slope, even if the numbers involved look positive at first glance.

Where this shows up next

Once graphing a single line feels solid, the natural next step is solving two equations at once graphically — finding where two lines cross — which is exactly what a system of equations asks you to do.

Continue Your Algebra Journey

Mastered graphing linear equations? Move on to the next topic.

Graphing Linear Equations Worksheet

Free printable practice: 26 problems with a full answer key and graphs.

Systems of Equations

Solve two equations at once.

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