Slope and Linear Equations: How to Find Slope and Write y = mx + b

What is slope?

Slope measures how steep a line is — how fast the $y$-value changes as the $x$-value changes. The classic definition is

where rise is the vertical change and run is the horizontal change between any two points on the line.

We almost always call slope $m$. A bigger $|m|$ means a steeper line; the sign of $m$ tells you the direction:

• $m > 0$: the line goes up from left to right.
• $m < 0$: the line goes down from left to right.
• $m = 0$: the line is horizontal ($y$ never changes).
• $m$ undefined: the line is vertical (run is zero, so we cannot divide).

The slope formula between two points

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope is

The numerator is the rise; the denominator is the run. The order of the points does not matter as long as you are consistent on top and bottom.

Example 1. Find the slope of the line through $(2, 3)$ and $(5, 9)$.

The line rises $2$ units for every $1$ unit you move right.

Example 2. Find the slope of the line through $(1, 4)$ and $(6, -1)$.

A slope of $-1$ means the line falls $1$ unit for every $1$ unit you move right.

Slope-intercept form: $y = mx + b$

The most useful form of a linear equation is slope-intercept form:

where:

• $m$ is the slope, and
• $b$ is the $y$-intercept — the $y$-value where the line crosses the $y$-axis (the point $(0, b)$).

Once a line is in this form, you can read off both pieces of information instantly.

Example 3. Identify the slope and $y$-intercept of $y = 3x - 2$.

The line has slope $m = 3$ and crosses the $y$-axis at $(0, -2)$, so $b = -2$.

Writing a line from two points

Given two points, you can produce the slope-intercept equation in three steps:

1. Compute the slope $m$ using the formula above.
2. Plug one point into $y = mx + b$ and solve for $b$.
3. Write the equation $y = mx + b$ with the values you found.

Example 4. Find the equation of the line through $(2, 5)$ and $(6, 13)$.

1. Slope: $m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = 2$.
2. Plug $(2, 5)$ into $y = 2x + b$: $5 = 2(2) + b$, so $b = 1$.
3. The line is $y = 2x + 1$.

You can sanity-check by plugging in the second point: $y = 2(6) + 1 = 13$. ✓

Graphing a line from $y = mx + b$

When the equation is already in slope-intercept form, graphing is fast:

1. Plot the $y$-intercept $(0, b)$.
2. Use the slope as rise/run from that point to find a second point.
3. Draw the line through the two points.

Example 5. Graph $y = \dfrac{1}{2}x + 3$.

• Start at $(0, 3)$.
• Slope $\dfrac{1}{2}$ means "rise $1$, run $2$." So the next point is $(2, 4)$.
• Draw the line through $(0, 3)$ and $(2, 4)$.

For a negative slope, you can either go down-right or up-left — both produce the same line.

Other forms you will see

You may also see lines written in these forms; each is just $y = mx + b$ in disguise.

• Point-slope form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is any point on the line. Useful when you know a point and a slope but not yet $b$.
• Standard form: $Ax + By = C$, with $A$, $B$, $C$ integers. To convert to slope-intercept form, solve for $y$.

Example 6. Convert $4x + 2y = 10$ to slope-intercept form.

Solve for $y$:

So the slope is $-2$ and the $y$-intercept is $(0, 5)$.

Common mistakes

• Subtracting in different orders top and bottom. $\dfrac{y_2 - y_1}{x_2 - x_1}$ — both subtractions must use the same order, or the sign of the slope flips.
• Reading the $y$-intercept off a wrong form. $b$ is the constant term in $y = mx + b$, not in $Ax + By = C$. Always solve for $y$ first.
• Mixing up rise and run. Rise is the change in $y$ (up/down); run is the change in $x$ (left/right). On the formula, rise is the numerator.
• Calling a vertical line "slope $0$." Vertical lines have undefined slope (the run is zero). Horizontal lines are the ones with slope $0$.