Answer:
<em>m [ slope ] = 8/3</em>
Step-by-step explanation:
What is the slope of the line that passes through the points (5, 1) and (2, -7)?
m = ∆y / ∆x
m = y₂ – y₁ / x₂ – x₁
| (5, 1) → (x₁, y₁) | (2, -7) → (x₂, y₂) |
[each coordinate will correspond (directly go to) to the variables shown]
m = (-7) – (1) / (2) – (5)
m = -8 / -3
m = 8 / 3
This means that this line has a rise of 8 and a run of 3.
Up 8 and right 3 proportionally, it would touch the opposite corners of a plane that was 8 by 3 units.
Slope can be thought of as the steepness of a line, the greater the slope, the greater the steepness.
These points are generally known as cartesian coordinates.
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Once you have the slope, you can back track from one of the coordinates to find the equation of the line.
Just multiply the slope by the x of either coordinate, make it opposite, and then add that to the y of that coordinate to get <em>b</em> [the y-intercept; where it crosses the y axis, x is always 0]
Since (2, -7) is closer to the y axis than (5, 1) it is more convenient.
In (2, -7), 2 is the x variable and -7 is the y variable.
-mx + y = (-8/3)(2) + (-7) = -16/3 + (-21/3)
21/3 is the same as 7 [so it will have a common denominator].
-16 + -21 / 3 = -37/3 = <em>b</em>
Thus this line has an equation of
y = mx + b
↓ ↓
y = 8/3x – 37/3.
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Or you can refer to point slope which is what is taught:
y – y₁ = m(x – x₁).
You can tell that there are some similarities to this and the slope formula.
m = ( y₂ – y₁ / x₂ – x₁ ) →
m × ( x₂ – x₁ ) =
( y₂ – y₁ / x₂ – x₁ ) × ( x₂ – x₁ )
[multiply the denominator by both sides]
m ( x₂ – x₁ ) = y₂ – y₁
[x₂, and y₂ can just be x and y since you only need the initial coordinate and slope, because the second coordinate can be anything along that line]
m ( x – x₁ ) = y – y₁ →
y – y₁ = m(x – x₁)
[symmetric property of equality]
if a = b, b = a.
