Algebra 1 · Pre-Algebra · Grades 8, 9

What Slope Is, and Why It Never Changes on a Line

Quick answer

Slope measures how steep a line is: the change in y divided by the change in x, often said as rise over run. Between the points (1, 2) and (4, 8) the slope is (8 - 2) divided by (4 - 1), which is 2. A positive slope rises left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has no slope at all.

What you'll learn

  • Find the slope of a line from a graph or from two points
  • Explain why slope is the same between any two points on a line
  • Interpret positive, negative, zero and undefined slope

What slope measures

Slope is a number describing how steep a line is, and which way it tilts. It compares how far the line goes up with how far it goes across:

slope=riserun=change in ychange in x\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}

A slope of 22 means: every time you move 11 across, the line climbs 22. A slope of 13\tfrac{1}{3} means it climbs only 11 for every 33 across, so it is much shallower.

The slope formula

Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on the line:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

The subscripts label the points; they are not exponents. y2y_2 means “the yy of the second point”, not ”yy squared”.

Why slope is the same everywhere on a line

Here is the question nobody asks out loud: if you pick two points near the left end of a line and I pick two near the right end, why do we get the same number?

It is not a coincidence, and it is not true of curves — pick two points on a parabola and the answer changes depending on where you look. Straightness is exactly what makes slope well-defined.

Draw the right triangle under your two points: horizontal leg the run, vertical leg the rise. Do the same for my two points. Both triangles have a right angle, and both have the line itself as their hypotenuse, so their acute angles match too.

Two triangles with the same three angles are similar: one is a scaled copy of the other. And in similar triangles, corresponding sides are in the same ratio. So

rise1run1=rise2run2\frac{\text{rise}_1}{\text{run}_1} = \frac{\text{rise}_2}{\text{run}_2}

Your triangle might be twice the size of mine, but doubling both the rise and the run leaves the fraction unchanged. That is why any two points give the same slope, and it is why a single number can describe a whole line.

The four kinds of slope

SlopeLine looks likeBecause
positiverises left to rightyy increases as xx increases
negativefalls left to rightyy decreases as xx increases
zerohorizontalno rise, so 0run=0\tfrac{0}{\text{run}} = 0
undefinedverticalno run, and dividing by zero is undefined

Worked examples

Common mistakes

Practice problems

  1. Find the slope through (0,1)(0, 1) and (2,7)(2, 7).

    Hint

    Rise over run: subtract the yy values on top, the xx values underneath.

    Answer

    m=3m = 3

    Full solution

    m=7−12−0=62=3m = \tfrac{7 - 1}{2 - 0} = \tfrac{6}{2} = 3.

    Check by swapping: 1−70−2=−6−2=3\tfrac{1 - 7}{0 - 2} = \tfrac{-6}{-2} = 3 ✓

  2. Find the slope through (1,5)(1, 5) and (4,−1)(4, -1).

    Answer

    m=−2m = -2

    Full solution

    m=−1−54−1=−63=−2m = \tfrac{-1 - 5}{4 - 1} = \tfrac{-6}{3} = -2. The line falls left to right.

  3. Find the slope through (−3,2)(-3, 2) and (5,2)(5, 2).

    Answer

    m=0m = 0

    Full solution

    m=2−25−(−3)=08=0m = \tfrac{2 - 2}{5 - (-3)} = \tfrac{0}{8} = 0. Equal heights mean a horizontal line.

  4. Find the slope through (4,−1)(4, -1) and (4,6)(4, 6).

    Hint

    Look at the denominator before you finish.

    Answer

    Undefined.

    Full solution

    m=6−(−1)4−4=70m = \tfrac{6 - (-1)}{4 - 4} = \tfrac{7}{0}, which is undefined. The line is vertical.

  5. Find the slope through (−2,−5)(-2, -5) and (2,3)(2, 3).

    Answer

    m=2m = 2

    Full solution

    m=3−(−5)2−(−2)=84=2m = \tfrac{3 - (-5)}{2 - (-2)} = \tfrac{8}{4} = 2.

  6. A line has slope 34\tfrac{3}{4} and passes through (0,1)(0, 1). Starting there, which point do you reach by moving one full rise and run?

    Hint

    A slope of 34\tfrac{3}{4} means up 33 for every 44 across.

    Answer

    (4,4)(4, 4)

    Full solution

    From (0,1)(0, 1), run 44 to the right and rise 33 up, landing on (4,4)(4, 4).

    Check: 4−14−0=34\tfrac{4 - 1}{4 - 0} = \tfrac{3}{4} ✓

  7. Do (1,3)(1, 3), (3,7)(3, 7) and (6,13)(6, 13) lie on one straight line?

    Hint

    If they are collinear, every pair gives the same slope.

    Answer

    Yes.

    Full solution

    From (1,3)(1,3) to (3,7)(3,7): m=42=2m = \tfrac{4}{2} = 2.

    From (3,7)(3,7) to (6,13)(6,13): m=63=2m = \tfrac{6}{3} = 2.

    From (1,3)(1,3) to (6,13)(6,13): m=105=2m = \tfrac{10}{5} = 2.

    All three agree, so the points are collinear. Had any pair disagreed, no straight line could pass through all three.

  8. A phone plan charges a fixed monthly fee plus a rate per gigabyte. Using 2 GB costs $35 and using 5 GB costs $50. What is the cost per gigabyte, and what does it mean?

    Hint

    Treat gigabytes as xx and cost as yy. The slope is the rate.

    Answer

    $5 per gigabyte.

    Full solution

    m=50−355−2=153=5m = \tfrac{50 - 35}{5 - 2} = \tfrac{15}{3} = 5.

    The slope is 55 dollars per gigabyte — each extra gigabyte adds $5 to the bill. The fixed fee is not the slope; it is where the line meets the vertical axis.

Frequently asked questions

Does it matter which point I call the first one?

No, as long as you are consistent. Subtracting in the other order flips the sign of both the top and the bottom of the fraction, and those two sign changes cancel. What breaks it is mixing the orders, using one point first on top and second on the bottom.

What is the difference between zero slope and undefined slope?

A horizontal line has slope 0, because y never changes so the top of the fraction is 0. A vertical line has undefined slope, because x never changes so the bottom is 0, and dividing by zero is not defined.

What to learn next

Key terms in this lesson

Linear equation
A linear equation is one whose graph is a straight line. Every variable appears to the first power only, so there are no squares, roots, or variables in a denominator.
Slope
Slope measures how steep a line is: the change in y divided by the change in x. A slope of 2 means the line climbs 2 units for every 1 unit across.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.8.EE.B.6Expressions and EquationsUse similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
  • CCSS.MATH.CONTENT.8.EE.B.5Expressions and EquationsGraph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
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