Question

How do you write an equation of a line going through (-5,-5), and (-2,3)?

Answer 1

`color(purple)(y=8/3x+25/3)`
Explanation given in detail to demonstrate methods. Shortcuts would reduce the method considerably!

Explanation:

Let the point be `P_1->(x_1,y_1)-> (-5,-5)`

Let the point be `P_2->(x_2,y_2)->(-2,3)`

Let the gradient (slope) be `m->("change in y-axis")/("change in x-axis")`

Let the constant be `c`

The standardised equation for a strait line graph is:

`color(white)("xxxxxxxxxxxxxxxxxx")y=mx+c` ..................(1)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find the slope (gradient)")`

`m=(y_2-y_1)/(x_2-x_1) = (3-(-5))/(-2-(-5)) =8/3`

Substitute for m in equation (1) giving:

`color(blue)(y=8/3x+c)`.......................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find the value of "c)`

At this stage if we substitute known value for `x` and `y` we end up with only one unknown. Which is c. Thus solvable.

Substitute `P_2->(x_2,y_2)->(-2,3)` into equation (2) giving:

`3=8/3(-2)+c`

`color(brown)(3=-16/3+c)`

Add `color(green)(16/3)` to both sides

`color(brown)(3 color(green)(+16/3 )=-16/3color(green)(+16/3 )+c)`

`25/3=0+c`

`color(blue)(c=25/3)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute for `c` in equation (2) giving:

`color(purple)(y=8/3x+25/3)`