Linear equations produce straight-line graphs and are the foundation of algebra. Understanding how to graph y = mx + b — and what each part means — is an essential skill for students at every level. This guide uses the free graphing calculator so you can try every example instantly.
The Slope-Intercept Form: y = mx + b
Every linear equation can be written as:
- m = the slope (steepness and direction of the line)
- b = the y-intercept (where the line crosses the y-axis)
Try it: Basic linear graph
Open the calculator and enter:
y = 2x + 1— slope 2, crosses y-axis at (0, 1)y = -x + 3— negative slope (goes down), crosses at (0, 3)
💡 A positive slope goes up from left to right. A negative slope goes down. A slope of 0 is a horizontal line.
Understanding Slope
Slope measures how much y changes for each 1-unit increase in x. It is calculated as:
m = (y₂ − y₁) / (x₂ − x₁) = rise / run
y = 3x— steep upward slope (rises 3 for every 1 right)y = 0.5x— gentle slopey = -2x— steep downward slopey = 5— horizontal line, slope = 0
Finding the X-Intercept
The x-intercept is where the line crosses the x-axis (y = 0). To find it algebraically, set y = 0 and solve for x.
For y = 2x + 4: set 0 = 2x + 4, so x = -2. The x-intercept is (-2, 0).
On the graphing calculator, root markers are shown automatically where the line crosses the x-axis. Hover over the marker to see the exact coordinate.
Parallel and Perpendicular Lines
Parallel lines
Parallel lines have the same slope but different y-intercepts. They never intersect. Try:
y = 2x + 1y = 2x - 3
Perpendicular lines
Perpendicular lines meet at a right angle. Their slopes are negative reciprocals: if one slope is m, the other is −1/m. Try:
y = 2x(slope = 2)y = -0.5x(slope = −1/2)
Systems of Linear Equations
A system of two linear equations has a solution where the lines intersect. Graph both lines to find the solution visually.
Enter both equations:
y = x + 2y = -x + 6
The calculator marks intersection points automatically. Hover over the intersection dot to read the exact solution (x = 2, y = 4 in this case).
Using Sliders to Explore
Type y = m*x + b in the calculator. Sliders for m and b appear automatically. Drag them to explore how changing slope and intercept affects the line in real time — a great way to build intuition.
Standard Form: Ax + By = C
If you have an equation in standard form like 2x + 3y = 12, rearrange it to slope-intercept form first:y = (-2/3)x + 4.
Or just enter it as 2*x + 3*y = 12 directly — the calculator can handle it.
- Graph
y = 3x - 2. What is the y-intercept? Where does it cross the x-axis? - Graph
y = x + 1andy = -x + 5. Where do they intersect? - Find a line parallel to
y = 4x + 1that passes through (0, −3).