y = mx + b: Read Slope & Y-Intercept, Plot a Line
y = mx + b: Read Slope & Y-Intercept, Plot a Line
TL;DR
In the equation y=mx+b, the number m is the slope and the number b is the y-intercept. This article shows how to read m and b, plot the line, write the equation from points or a graph, and rearrange any linear equation into this form.
What Do m and b Mean in y = mx + b?
In the equation y=mx+b, every line is described by two numbers:
- m is the slope, measuring how steep the line is, as rise over run: how many units y changes for every one unit x moves right. A positive m slopes up to the right; a negative m slopes down; a larger |m| is steeper.
- b is the y-intercept, where the line crosses the y-axis at the point (0,b). Setting x=0 in the equation gives y=b.
For example, in y=2x+3, the slope is m=2 and the y-intercept is b=3.
How Do You Read the Slope and Y-Intercept From the Equation?
When the equation is written as y=mx+b, you read:
- The coefficient of x is the slope m.
- The constant term is the y-intercept b.
| Equation | Slope m | Y-intercept b | Crosses y-axis at |
|---|---|---|---|
| y=4x+1 | 4 | 1 | (0,1) |
| y=-3x+5 | -3 | 5 | (0,5) |
| y=1/2x-2 | 1/2 | -2 | (0,-2) |
| y=x | 1 | 0 | (0,0) |
| y=7 | 0 | 7 | (0,7) |
How Do You Plot a Line From y = mx + b?
- Plot the y-intercept first at (0,b).
- Use the slope to find the second point. Write m as rise/run. For m=2, go 1 unit right and 2 units up from (0,3) to find (1,5).
- Draw the line through the two points.
How Do You Write the Equation From Two Points or a Graph?
- Find the slope using the points (x1,y1) and (x2,y2): m=(y2-y1)/(x2-x1).
- Find b by substituting one point into y=mx+b and solving for b.
How Do You Rearrange Any Line Into y = mx + b?
To convert from standard form Ax+By=C to slope-intercept form:
- Isolate y: 4y=-3x+12 4y=-3x+12 y=-3/4x+3
Now slope m=-3/4 and y-intercept b=3.
Examples of y = mx + b
Example 1: Slope and y-intercept of y=5x−2
m=5, b=−2.
Example 2: Slope and y-intercept of 2x+y=6
Rearranging gives y=−2x+6, thus m=−2, b=6.
Example 3: Plotting y=−x+4
Start at (0,4), slope m=−1, resulting in the line.
Example 4: Write the equation through (0,−1) and (3,5)
m=2, b=−1: y=2x−1.
Example 5: Rearranging 6x−2y=8
gives y=3x−4, thus m=3, b=−4.
Example 6: Line through (0,2) and (4,0)
y=−1/2x+2.
Why y = mx + b Matters Beyond the Page
- A starting value plus a steady rate.
- Reading trends.
- The bridge to calculus.
- Descartes' gift of connecting geometry and algebra.
Mistakes to Avoid on y = mx + b
- Reading before solving for y.
- Mixing up m and b.
- Mishandling the direction for negative slopes.
Conclusion
In y=mx+b, m is the slope and b is the y-intercept. Read m as the coefficient of x and b as the constant, but only when y is solved for.
Practice Problems
- State the slope and y-intercept of y=−2x+7.
- Rearrange 4x+2y=10 into slope-intercept form and state m and b.
- Write the equation of the line through (0,4) and (2,0).