Question

What is the equation of the line passing through `(4,8)` and `(-9,3)`?

Answer 1

point-slope form:
`y - 8 = frac{5}{13}(x-4)`
or
`y - 3 = frac{5}{13}(x+9)`

slope-intercept form:
`y = frac(5)(13)x + frac(84)(13)`

standard form:
`-5x + 13y = 84`

Explanation:

Method 1:
Use point slope form
which is `y - y_1 = m(x - x_1)`
when given a point `(x_1, y_1)` and the slope `m`
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In this case, we should first find the slope between the two given points.
This is given by the equation:
`m = frac{y_2 - y_1}{x_2 - x_1}`
when given the points `(x_1,y_1)` and `(x_2, y_2)`
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For `(x_1,y_1) = (4,8)` and `(x_2,y_2) = (-9,3)`
By plugging what we know into the slope equation, we can get:
`m = frac{3-8}{-9-4} = frac{-5}{-13} = frac{5}{13}`
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from here we can plug in either point and get:
`y - 8 = frac{5}{13}(x-4)`
or
`y - 3 = frac{5}{13}(x+9)`

Method 2:
Use slope intercept form
which is `y = mx + b`
when `m` is the slope and `b` is the y-intercept
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We can find the slope between the two given points using the same steps as above
and get `m= frac{5}{13}`
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but this time when we plug in, we will still be missing the `b` or y-intercept
to find the y-intercept, we need to temporarily plug in one of the given points in for `(x,y)` and solve for b
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so
`y= frac{5}{13}x + b`
if we plug in `(x,y)=(4,8)`
we would get:
`8 = frac(5)(13)(4) + b`
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solving for `b` would get us
`8 = frac{20}{13} + b`
`b = 84/13 or 6 frac(6)(13)`
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so your equation would be
`y = frac(5)(13)x + frac(84)(13)`

another form your equation could be in can be standard form where only the variables are on one side
`ax + by = c`
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you can get you equation into this form by multiplying both sides of the slope intercept equation by 13
to get `13y = 5x + 84`
then subtract `5x` from both sides
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so your standard form equation would be
`-5x + 13y = 84`