Question

How do you write an equation in point slope form that passes through (1,4) and(4,9)?

Answer 1

`y=color(green)(5/3)x+color(purple)(7/3)`

Explanation:

We want this in the form `y=color(green)(m)x+color(purple)(c)`, where `m` is the gradient, and `c` is the y-intercept.

Gradient is the change in y over the change in x, or the rise over run; however you like to think about it.

`color(green)m=(color(red)(y_2)-color(blue)(y_1))/(color(red)(x_2)-color(blue)(x_1))`

So for our points `(color(blue)(1,4))` and `(color(red)(4,9))`

`m=(color(red)9-color(blue)4)/(color(red)4-color(blue)1)`

`=5/3`

Now, by rearranging our equation for gradient, we can show that:

`y-color(blue)(y_1)=color(green)(m)(x-color(blue)(x_1))`

Here, `x` and `y` are general points on the line - we've just replaced `(color(red)(x_2,y_2))` with `(color(red)(x,y))`

Since we know `color(green)m`, and have a set of co-ordinates to sub in as `color(blue)((x_1, y_1)`, we can sub values in, then re-arrange to get to the `y=color(green)(m)x+color(purple)(c)` form.

`y-color(blue)(4)=color(green)(5/3)(x-color(blue)(1))`

`y-4=5/3x-5/3`
`y=5/3x-5/3+4`
`y=color(green)(5/3)x+color(purple)(7/3)`
as required.

---

An alternative method, after finding the gradient, is to substitute your gradient and a pair of points into `y=color(green)(m)x+color(purple)(c)`, rearrange to find `c`, then go back afterwards and write the equation. If you've been taught that way, its fine! But personally I prefer substituting straight into `y-color(blue)(y_1)=color(green)(m)(x-color(blue)(x_1))` since you don't have to go back afterwards. Up to you which method you use.

Answer 2

`y-4=5/3(x-1)`

Explanation:

`"the equation of a line in "color(blue)"point-slope form"` is.

`•color(white)(x)y-y_1=m(x-x_1)`

`"where m is the slope and "(x_1,y_1)" a point on the line"`

`"to calculate m use the "color(blue)"gradient formula"`

`•color(white)(x)m=(y_2-y_1)/(x_2-x_1)`

`"let "(x_1,y_1)=(1,4)" and "(x_2,y_2)=(4,9)`

`m=(9-4)/(4-1)=5/3`

`"use either of the 2 given points as point on line"`

`"using "(1,4)" then"`

`y-4=5/3(x-1)larrcolor(red)"in point-slope form"`