Every straight line on a graph can be written as one short equation. It is the most important equation in GED algebra — and once you know what its two letters mean, you can graph almost any line in about thirty seconds.
y = mx + b
m = slopeb = y-intercept
The main point: The y-intercept is your starting point on the graph. The slope tells you where to go next. That's the whole skill — find b, plot it, then use rise over run.
Part 1A — Slope
m = slope
How steep the line is, and which way it tilts.
Slope = riserun
Positive slope → line rises left to right. Negative slope → line falls left to right.
Part 1B — Y-Intercept
b = y-intercept
Where the line crosses the y-axis — your starting point.
Always at the point (0, b).
A positive b crosses above zero.
A negative b crosses below zero.
Watch the minus signs. In y = −x + 4 the slope is −1, not 1. In y = 2x − 7 the y-intercept is −7, not 7.
Whole-number slopes are fractions in disguise. 2 = 21 | −3 = −31 | 5 = 51
So a slope of 2 means "up 2, right 1." A slope of −3 means "down 3, right 1."
Part 3 — How to Graph a Line from y = mx + b
Plot b. Then use slope (rise over run) to step to a second point.
1. Identify b in the equation. Plot the point (0, b) on the y-axis. 2. Identify m. Write it as a fraction riserun. 3. From your starting point, go rise up or down, then run right. 4. Plot the second point. Draw a straight line through both.
Part 4 — Writing an Equation from a Graph
Read b off the y-axis. Count rise/run between two clean points. Plug into y = mx + b.
Step 1. Where does the line cross the y-axis? That number is b. Step 2. Pick two points where the line passes through exact grid corners. Count up/down (rise) and right (run). That fraction is m. Step 3. Write y = mx + b.
Common mistakes to avoid: • Mixing up slope and y-intercept (the number with x is always slope).
• Plotting b on the x-axis instead of the y-axis.
• Reversing rise and run (rise is on top of the fraction).
• Losing a negative sign — negative slope means the line falls.
• Forgetting that a negative y-intercept crosses below the x-axis.
🔢 Worked Examples
Example 1 — Graph y = 23x + 1
Identify the y-intercept and the slope, then plot two points and draw the line.
b = 1 → start at the point (0, 1)
m = 23 → rise 2, run 3
From (0, 1), go up 2 then right 3 → land at (3, 3)
Draw a line through (0, 1) and (3, 3)
Plot (0, 1) first. Step up 2, then right 3. Draw the line.
Example 2 — Graph y = −x + 4
A negative slope means the line falls from left to right. The y-intercept is positive.
b = 4 → start at (0, 4)
m = −1 = −11 → down 1, right 1
From (0, 4), go down 1 then right 1 → (1, 3)
Draw the line through (0, 4) and (1, 3)
Negative slope ⇒ the line falls. Slope of −1 means down 1 for every 1 right.
Example 3 — Graph y = 32x − 1
Negative y-intercepts start below the x-axis. Fractional slopes use the bottom number for run.