Why slope is the same everywhere on a line
Tonight's idea
Take a straight line and two points on it. Travel from one to the other: so far across (the run), then so far up (the rise). Rise ÷ run is the slope, written \(m\).
Different points give a different triangle. Why the same fraction? Both triangles have a right angle and meet the line at the same angle, since their horizontal legs are parallel. Same angles, same shape: the big one is a scaled copy of the small one. Scaling multiplies rise and run by the same factor, so the ratio is untouched. Slope belongs to the line, not to the points you picked.
\(m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}\) for any two points \((x_1, y_1)\), \((x_2, y_2)\) on the line.
That is why \(y = mx + b\) works. Say the line crosses the \(y\)-axis at \((0, b)\) with slope \(m\). From there to any other point \((x, y)\) on it, the rise is \(y - b\) and the run is \(x\), so \(\dfrac{y - b}{x} = m\), giving \(y = mx + b\). Points on the line obey this; points off it do not.
A falling line has a negative rise, so a negative slope. A flat line has slope 0. A vertical line has run 0, and you cannot divide by 0, so it has no slope.
Watch one: the line through \((1, 3)\) and \((4, 9)\)
Run \(4 - 1 = 3\), rise \(9 - 3 = 6\), so \(m = 6/3 = 2\). Up 2 for every 1 across.
Where is the point on this line with \(x = -2\)? From \((1, 3)\) the run is \(-2 - 1 = -3\), so the rise is \(2 \times (-3) = -6\). The point is \((-2, -3)\).
The equation: \(y = 2x + b\), and \((1, 3)\) is on the line, so \(3 = 2 + b\) and \(b = 1\). Check \((4, 9)\): \(2(4) + 1 = 9\). Check \((-2, -3)\): \(2(-2) + 1 = -3\).
Drag two points along a line and watch the triangle change size but not shape: mathdetours.com/demos/lines.html.
Your turn
- Find the slope of the line through each pair of points: (a) \((2, 5)\) and \((6, 13)\); (b) \((-1, 4)\) and \((3, -2)\); (c) \((0, 7)\) and \((5, 7)\).
- A line has slope \(\tfrac34\) and passes through \((4, 1)\). (a) Give one more point on it to the right and one to the left. (b) Is \((12, 7)\) on the line? Is \((10, 6)\)?
- (a) Are \((1, 2)\), \((3, 8)\) and \((7, 20)\) on one straight line? Use slopes, not a drawing. (b) Same question for \((1, 2)\), \((3, 8)\) and \((6, 16)\). (c) Write the equation of the line in (a).
- A regulation says a ramp may rise at most 1 unit for every 12 along the ground. (a) A door is 75 cm above the ground. What is the shortest legal horizontal length for a straight ramp up to it? (b) An inspector measures a 150 cm stretch of a long straight ramp: it rises 11 cm. Is the ramp legal? Why is one stretch enough to decide?
Stretch: turn the triangle
A line has slope \(\tfrac34\): its triangle goes 4 across and 3 up. Rotate the triangle a quarter turn about one corner. The new hypotenuse is perpendicular to the old line. Read off rise and run; find the slope. Repeat for a general slope \(m\) that is not 0, and show the two slopes multiply to \(-1\).
Answers
- (a) \(8/4 = 2\). (b) \(-6/4 = -\tfrac32\). (c) \(0/5 = 0\): the line is horizontal.
- (a) Add 4 to \(x\) and 3 to \(y\), or subtract both: \((8, 4)\) and \((0, -2)\), for example. (b) From \((4, 1)\) to \((12, 7)\): rise 6, run 8, and \(6/8 = \tfrac34\). Yes. To \((10, 6)\): \(5/6 \neq \tfrac34\). No.
- (a) \(6/2 = 3\) and \(12/4 = 3\): the same slope along both stretches, so yes. (b) \(6/2 = 3\) but \(14/5 \neq 3\): no. (c) \(y = 3x + b\) with \(2 = 3 + b\), so \(y = 3x - 1\). Check: \(3(7) - 1 = 20\).
- (a) Slope at most \(\tfrac{1}{12}\), so the run is at least \(12 \times 75 = 900\) cm, which is 9 m. (b) \(11/150 \approx 0.073\), under \(1/12 \approx 0.083\): legal. A straight ramp has one slope, so any stretch of it has the same rise-to-run ratio as the whole.
Stretch. After a quarter turn the leg that went 4 across now goes 4 up, and the leg that went 3 up now goes 3 back: rise 4, run \(-3\), slope \(-\tfrac43\). In general a triangle with run 1 and rise \(m\) turns into one with rise 1 and run \(-m\), slope \(\dfrac{1}{-m} = -\dfrac{1}{m}\). The product is \(m \times \left(-\dfrac1m\right) = -1\).