How To Find Slope

Slope measures how fast a line changes: divide the change in $y$ by the change in $x$. If you know two points, use

as long as $x_2 \ne x_1$. That is the same idea as rise over run, how much the line goes up or down compared with how far it goes right. A positive slope means the line rises from left to right, a negative slope means it falls, and a slope of $0$ means it is horizontal. If $x_2 - x_1 = 0$, the line is vertical and the slope is undefined, because the formula would require division by $0$.

When To Use Each Approach

Slope is a rate of change, comparing how much $y$ changes with how much $x$ changes, which is why it shows up in algebra, graphs, and data tables. Pick your method by what you are handed:

• Two coordinates given directly: use the slope formula.
• A drawn line: count rise and run off the grid.
• A table of values: take the change in $y$ over the change in $x$ between two rows, but only when the rate is constant.

The Core Procedure From Two Points

Keep 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.

Reverse both subtraction orders and the slope is unchanged; reverse only one and the sign flips.

The Whole Procedure On One Example

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

Substitute the coordinates in the same order:

The slope is $2$: every time $x$ increases by $1$, $y$ increases by $2$. Read as rise over run, from $(2, 3)$ to $(5, 9)$ the line goes up $6$ and right $3$, so the slope is $6/3 = 2$.

The Same Procedure On 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 $2$:

If you move down $3$ and right $1$:

Using grid intersection points helps avoid counting mistakes.

The Same Procedure From A Table

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

If different row pairs give the same value, the relationship is linear and that constant is the slope. For example, if $x$ increases from $1$ to $3$ while $y$ increases from $4$ to $10$:

Where Each Step Gets Stuck, And How To Check

• At step 3 — different subtraction orders. If you use $y_2 - y_1$, you must also use $x_2 - x_1$. Self-check: did both subtractions go the same direction?
• At the divide step — calling a vertical line's slope $0$. If two points share an $x$-value, the denominator is $0$, so the slope is undefined, not zero. Self-check: is the denominator nonzero before you divide?
• From a table — assuming a slope exists. A table has one slope only if the rate of change stays constant. Self-check: do at least two row pairs agree?

When Slope Is Used

Slope appears whenever you describe how one quantity changes against another: graphing lines, writing linear equations, physics formulas with constant rates, and tables that follow a linear pattern.

For a quick run-through, find the slope between $(1, -2)$ and $(4, 7)$. Write the subtraction step before you simplify, confirm the denominator is nonzero, then decide whether the line rises or falls as $x$ increases.