Question

How do you write an equation in standard form given point (-3,-1) and (2,5)?

Answer 1

`6x-5y=-13`

Explanation:

The equation of a line in `color(blue)"standard form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(Ax+By=C)color(white)(2/2)|)))`
where A is a positive integer and B, C are integers.

Begin by expressing the equation in `color(blue)"point-slope form"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(y-y_1=m(x-x_1))color(white)(2/2)|)))`
where m represents the slope and `(x_1,y_1)" a point on the line"`

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

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

The 2 points here are (-3 ,-1) and (2 ,5)

let `(x_1,y_1)=(-3,-1)" and " (x_2,y_2)=(2,5)`

`rArrm=(5-(-1))/(2-(-3))=6/5`

Either of the 2 given points can be used for `(x_1,y_1)`

`"Using "m=6/5" and " (x_1,y_1)=(2,5)` we can establish the equation.

`rArry-5=6/5(x-2)larrcolor(red)" in point-slope form"`

distribute bracket and rearrange into standard form.

`y-5=6/5x-12/5`

multiply ALL terms on both sides by 5

`rArr5y-25=6x-12`

`rArr6x-5y=-13larrcolor(red)" in standard form"`