Question

What is the equation of the line passing through (-3,-2 ) and (1, -5)?

Answer 1

`y=-3/4x-17/4`

Explanation:

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

`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 ,-2) and (1 ,-5)

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

`rArrm=(-5-(-2))/(1-(-3))=(-3)/4=-3/4`

We can use either of the points (-3 ,-2), (1 ,-5) as the point on the line since the line passes through both of them.

`"Using " (x_1,y_1)=(1,-5)" and "m=-3/4`

Substitute these values into the equation.

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

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

Distributing and simplifying gives an alternative version of the equation.

`y+5=-3/4x+3/4`

`rArry=-3/4x+3/4-5`

`rArry=-3/4x-17/4larrcolor(red)"in slope-intercept form"`