# y = mx + b: Read Slope & Y-Intercept, Plot a Line

[Geometry](/content/tag/geometry/index.html)

## 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?

1. **Plot the y-intercept first** at (0,b).
2. **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).
3. **Draw the line** through the two points.

## How Do You Write the Equation From Two Points or a Graph?

1. **Find the slope** using the points (x1,y1) and (x2,y2): m=(y2-y1)/(x2-x1).
2. **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:
1. 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

1. State the slope and y-intercept of y=−2x+7.
2. Rearrange 4x+2y=10 into slope-intercept form and state m and b.
3. Write the equation of the line through (0,4) and (2,0).