The slope of the line through (9, −10) and (14, 5) is 3. Slope equals rise over run: the change in y is 15, the change in x is 5, and 15 divided by 5 simplifies to 3, so the line rises three vertical units for every one horizontal unit across (9, −10) and (14, 5). You can verify the result from either endpoint, because the slope formula returns 3 whether you subtract the first point from the second or the reverse.
How the Slope Formula Turns Two Points Into One Number
The slope formula is m = (y₂ − y₁) ÷ (x₂ − x₁). The letter m is the conventional symbol; it does not stand for anything you need to compute. The numerator measures vertical change, and the denominator measures horizontal change. Dividing one by the other expresses steepness as a single rate.
Assign your points deliberately. Label (9, −10) as (x₁, y₁) and (14, 5) as (x₂, y₂). Then y₂ − y₁ becomes 5 − (−10), and x₂ − x₁ becomes 14 − 9. The subtraction of a negative number is the step where most errors happen.
Work the arithmetic in two separate lines. The numerator is 5 + 10 = 15, because subtracting −10 is the same as adding 10. The denominator is 14 − 9 = 5. The fraction is 15/5, which simplifies to 3 because 5 divides 15 exactly three times.
The result is a positive integer, and that sign carries information. A positive slope means the line climbs as you read left to right. Both coordinates of the second point are larger than those of the first, so the climb is guaranteed before you compute anything.
Try the subtraction in reverse to test robustness. Let (14, 5) be the first point and (9, −10) the second. The numerator becomes −10 − 5 = −15, and the denominator becomes 9 − 14 = −5. The fraction −15/−5 also simplifies to 3, because a negative divided by a negative is positive.
That reversal is a useful self-check, not a trick. Consistent point order is the only rule, and swapping both coordinates together never changes the quotient. Mixing orders between numerator and denominator is what produces a wrong sign.
In plain terms, the formula is a division problem about direction. You ask how much the line travels vertically for each unit it travels horizontally. For these two points, the answer is three units up for every one unit right.
What the Number 3 Means on a Coordinate Grid
The slope 3 is a slope of 3/1 in fractional form. From (9, −10), move right 1 unit to x = 10 and up 3 units to y = −7. That lands on (10, −7), which satisfies the same line equation. Repeat the move four more times and you arrive at (14, 5).
This stepping behavior is why slope is called a rate of change. Each horizontal step of 1 costs a vertical step of 3. The line is steep relative to a slope of 1, which would rise at exactly 45 degrees on a grid with equal axis scaling.
The angle is measurable. The inverse tangent of 3 is roughly 71.6 degrees above horizontal. The inverse tangent of 1 is 45 degrees, and the inverse tangent of 0.5 is roughly 26.6 degrees. Slope values above 1 describe lines that climb faster than they run.
The line's full equation follows from the same data. Using point-slope form with (14, 5), you get y − 5 = 3(x − 14). Expanded, that is y = 3x − 37. Substituting x = 9 returns y = −10, which confirms both points sit on the line.
A quick sanity check uses the average rate of change. Over the interval from x = 9 to x = 14, y changes by 15 while x changes by 5, giving 3 per unit. Dividing total change by total run works for any straight line because the rate never varies.
Interpretations vary by context even when the number is identical. A slope of 3 could mean 3 dollars per unit sold, 3 meters of elevation per meter walked, or 3 degrees of temperature change per hour. The geometry is fixed; the meaning comes from the axis labels.
Comparing This Line With Nearby Alternatives
Slope questions cluster around a few common point pairs, and comparing them shows how sensitive the answer is to signs. The table below uses the same formula throughout. All values are exact rational numbers, not rounded decimals.
| Point pair | Numerator (y₂ − y₁) | Denominator (x₂ − x₁) | Slope (simplest form) | Line behavior |
|---|---|---|---|---|
| (9, −10) and (14, 5) | 15 | 5 | 3 | Steep positive climb |
| (9, 10) and (14, 5) | −5 | 5 | −1 | Downward at 45 degrees |
| (9, −10) and (14, −5) | 5 | 5 | 1 | Upward at 45 degrees |
| (9, −10) and (9, 5) | 15 | 0 | Undefined | Vertical line |
| (9, −10) and (14, −10) | 0 | 5 | 0 | Horizontal line |
| (14, 5) and (9, −10) | −15 | −5 | 3 | Same line, reversed order |
The table exposes a pattern. Sign changes in the numerator flip the slope's direction, and a zero denominator breaks the formula entirely. A zero numerator gives a flat line with slope 0, which is a defined number, not an undefined one.
The vertical case deserves care because it is not a number. Dividing by zero is undefined in arithmetic, and no finite slope describes a vertical line. Textbooks often write the slope as undefined or say it does not exist. Do not write infinity on a test unless your course explicitly allows it.
The repeated pair at the bottom shows that the calculation is order-independent when done correctly. Students who get −3 usually flipped only one of the two subtractions. Checking the sign against the visual direction of the points catches that error immediately.
Where Slope Calculations Go Wrong
Sign errors dominate. Subtracting −10 incorrectly as −10 instead of +10 turns the numerator into −5, and the slope becomes −1. That single slip changes a steep climb into a 45-degree descent, which contradicts the obvious fact that y increases from −10 to 5.
Order mixing is the second common failure. Using y₂ − y₁ in the numerator but x₁ − x₂ in the denominator reverses the sign of the whole fraction. The fix is mechanical: write the coordinates under the formula before substituting, so each position is filled consistently.
Simplification errors are subtler. A result like 15/5 must be reduced, and graders often require simplest form. Any fraction where the numerator and denominator share a factor larger than 1 is incomplete. Here the reduction is exact, leaving the integer 3.
Red-flag warnings worth committing to memory:
Subtracting a negative y-value without changing it to addition, which flips the sign. Swapping only one coordinate pair between numerator and denominator. Reporting a vertical line's slope as 0 instead of undefined. Leaving a fraction unreduced when the problem demands simplest form. Rounding a slope to a decimal when an exact fraction is available. Trusting a mental shortcut over a written substitution step.
Decimals are a related trap. Writing 3 as 3.0 is harmless, but writing 15/5 as 3.0000001 from a calculator display suggests false precision. Exact answers from integer coordinates are always rational, and simplest form is preferred unless the problem states otherwise.
Practical Steps for Any Two-Point Slope Problem
A repeatable procedure removes most of the risk. The steps below work for any pair of distinct points with different x-values. Each step is short enough to check in seconds.
Label the points as (x₁, y₁) and (x₂, y₂) in the order given, and write them on separate lines. Write the formula m = (y₂ − y₁)/(x₂ − x₁) before substituting any numbers. Substitute with parentheses around every negative value, then convert subtraction of a negative to addition. Compute numerator and denominator separately, keeping signs visible. Reduce the fraction by the greatest common factor, or state undefined if the denominator is 0. Verify by reversing the point order to confirm the same slope, and check that the sign matches the direction of the points.
For this problem, the steps produce 5 − (−10) in the numerator and 14 − 9 in the denominator. Those become 15 and 5, then 3 after reduction. The reversal check gives −15/−5, which also equals 3. The direction check passes because y rises as x rises.
If a course requires slope-intercept form, continue from the slope. Substitute either point into y = mx + b to solve for b. Using (14, 5) with m = 3 gives 5 = 42 + b, so b = −37 and the line is y = 3x − 37, matching the earlier derivation.
When the Slope Formula Stops Being Enough
The formula answers a narrow question: the constant rate of change between two distinct points with different x-values. It says nothing about what happens beyond those points, because two points define a line only under the assumption that the relationship is linear at all.
Vertical lines break the formula because the run is zero. Horizontal lines are fine and produce slope 0. Coincident points also break it, because both numerator and denominator become zero and the expression 0/0 is indeterminate rather than meaningful.
Real data often bends. A slope of 3 computed between two measured points describes only the average rate over that interval, not the instantaneous rate at either end. Curves can have a slope of 3 at one point and a different slope a short distance away.
Rounding introduces a separate limit. Coordinates reported to the nearest whole number give an exact slope only if the underlying values are exact. If the points are measurements with uncertainty, the slope inherits that uncertainty and should not be reported with more digits than the inputs justify.
The counterintuitive detail is that a large slope does not always mean a large change. Slope is a ratio, so a slope of 3 can come from a change of 3 over 1 or from a change of 3,000 over 1,000. The steepness on a scaled graph depends on how the axes are drawn, not on the raw magnitude of the changes.
The safest habit is to state the answer with its units and its interval. Saying the slope is 3 over the span from x = 9 to x = 14 is more honest than saying the slope is 3 everywhere. Two points constrain a line, not a trend, and they cannot distinguish a genuine steady rate from a coincidence between endpoints.