How To Find Slope

To find slope, divide the change in $y$ by the change in $x$. If you know two points, use the slope formula

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

as long as $x_2 \ne x_1$. This is the same idea as rise over run: how much the line goes up or down compared with how far it goes right.

Slope tells you how fast a line changes. A positive slope means the line rises from left to right, a negative slope means it falls, and a slope of $0$ means the line is horizontal.

If $x_2 - x_1 = 0$, the line is vertical. In that case the slope is undefined because the formula would require division by $0$.

What Slope Means

Slope is a rate of change. It compares how much $y$ changes with how much $x$ changes.

That is why slope shows up in algebra, graphs, and data tables. The same idea works anywhere a relationship changes at a constant rate.

How To Find Slope From Two Points

Use the same subtraction order in the numerator and denominator:

1. Pick the two points.
2. Subtract the $y$-values to get the change in $y$.
3. Subtract the $x$-values in the same order to get the change in $x$.
4. Divide.
5. Simplify if possible.

If you reverse both subtraction orders, the slope stays the same. If you reverse only one, the sign will be wrong.

Worked Example: Find The Slope Between Two Points

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

Start with the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substitute the coordinates in the same order:

$$m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$$

The slope is $2$. That means every time $x$ increases by $1$, $y$ increases by $2$.

You can also read this as rise over run. From $(2, 3)$ to $(5, 9)$, the line goes up $6$ and right $3$, so the slope is $6/3 = 2$.

How To Find Slope From A Graph

Pick two clear grid points on the line. Count the vertical change first, then the horizontal change.

If you move up $4$ and right $2$, the slope is

$$\frac{4}{2} = 2$$

If you move down $3$ and right $1$, the slope is

$$\frac{-3}{1} = -3$$

Using grid intersection points helps avoid counting mistakes.

How To Find Slope From A Table

A table gives a slope only when the rate of change is constant. Choose two rows and compute

$$\frac{\text{change in } y}{\text{change in } x}$$

If you get the same value from different row pairs, the relationship is linear and that constant value is the slope.

For example, if $x$ increases from $1$ to $3$ while $y$ increases from $4$ to $10$, then

$$m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$$

Common Mistakes When Finding Slope

One common mistake is subtracting in different orders. If you use $y_2 - y_1$, you must also use $x_2 - x_1$.

Another mistake is calling a vertical line's slope $0$. If two points have the same $x$-value, the denominator is $0$, so the slope is undefined.

A third mistake is assuming any table has a slope. A table has one slope only if the rate of change stays constant.

When Slope Is Used

Slope is used whenever you want to describe how one quantity changes compared with another. You see it in graphing lines, writing linear equations, physics formulas with constant rates, and data tables that follow a linear pattern.

Try Your Own Version

Find the slope between $(1, -2)$ and $(4, 7)$. Write the subtraction step before you simplify, then decide whether the line rises or falls as $x$ increases.

If you want one more case, try your own version with two new points and check whether the denominator stays nonzero before you divide.