How do you the equation of the line that passes through point (4,2) and (6,6)?

The slope of the line through `(4,2)` and `(6,6)`
is `color(green)(m)=(Deltay)/(Deltax)=(6-2)/(6-4)=4/2=color(green)(2)`

The slope-point form for a line
with slope `color(green)(m)`
through the point `(color(red)(a),color(blue)(b))`
is `y-color(blue)b=color(green)(m)(x-color(red)(a))`

Using `(color(red)4,color(blue)2)` as our point (we could have used either of the given points)
and the previously determined `color(green)(m=2)`

The slope-point form for the required line is
`color(white)("XXX")y-color(blue)(2)=color(green)(2)(x-color(red)4)`

This could easily be converted into slope-point form as
`color(white)("XXX")y=color(green)(2)x-6`

or into standard form as
`color(white)("XXX")x-2y=6`

The equation of a line in `color(blue)"slope-intercept form"` is

`color(red)(bar(ul(|color(white)(a/a)color(black)(y=mx+b)color(white)(a/a)|)))`
where m represents the slope and b, the y-intercept.

To calcuate m, use the `color(blue)"gradient formula"`

`color(red)(bar(ul(|color(white)(a/a)color(black)(m=(y_2-y_1)/(x_2-x_1))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (4 ,2) and (6 ,6)

let `(x_1,y_1)=(4,2)" and " (x_2,y_2)=(6,6)`

`rArrm=(6-2)/(6-4)=4/2=2`

Thus partial equation is : `y=2x+b`

To find b, substitute either of the 2 given points into the partial equation and solve for b.

Using (4 ,2) : `(2xx4)+b=2rArrb=2-8=-6`

`rArry=2x-6" is the equation of the line"`