Key takeaway: The slope formula finds the steepness of a line from any two of its points by dividing the difference of the y-coordinates by the difference of the x-coordinates.
Grade 88.EE.B.6Functions and Graphs
Definition of Slope Formula
The slope formula turns two points into a number. You do not need a graph: take the two coordinate pairs, subtract the y-values for the rise, subtract the x-values for the run, and divide. The result is the same no matter which two points of the line you pick.
One rule keeps it honest. Subtract in the same order on top and bottom. If the numerator starts with the second point’s y-value, the denominator must start with the second point’s x-value. Reversing both gives the same answer; reversing only one flips the sign. Pay special attention when coordinates are negative, since subtracting a negative number adds.
Formula and notation
Name the points A and B, with coordinates \((x_A, y_A)\) and \((x_B, y_B)\), where \(x_A \neq x_B\):
\[m = \frac{y_B – y_A}{x_B – x_A}\]
If the two x-values match, the denominator is 0 and the vertical line has an undefined value.
Key facts
- The top and bottom subtractions must follow the same order of points.
- A negative result means the line falls as you move right.
- Wrap negative coordinates in parentheses before subtracting.
- A zero on top means a horizontal line; a zero underneath means a vertical one.
Slope Formula example
Use the slope formula to find the steepness of the line through \(A(-2, 5)\) and \(B(4, -4)\).
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Rise: \(y_B – y_A = -4 – 5 = -9\).
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Run: \(x_B – x_A = 4 – (-2) = 6\).
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Divide: \(m = \dfrac{-9}{6} = -\dfrac{3}{2}\).
The value is \(-\dfrac{3}{2}\): the line falls 3 units for every 2 units to the right.
Common mistake
Watch out
Cleo computes \(\dfrac{-4 – 5}{-2 – 4} = \dfrac{-9}{-6} = \dfrac{3}{2}\) for the same two points.
Why it is wrong
She subtracted in opposite orders: B minus A on top but A minus B underneath. The sign flips, giving a positive result for a falling line. Use the same order twice: \(\dfrac{-4 – 5}{4 – (-2)} = -\dfrac{3}{2}\). A quick glance at the graph confirms the line goes down, and the slope formula must agree.
Related terms and lessons
- Grade 8 · Proportional Relationships and SlopeLessonWorksheetQuiz
- Algebra 1 · Linear Functions: Slope-Intercept, Point-Slope and Standard FormLessonWorksheetQuiz
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