Question

What is the equation of the line that passes through the points (2, 4) and (4,0)?

Answer 1

`y=-2x+8`

Explanation:

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=mx+b)color(white)(2/2)|)))`
where m represents the slope and b, the y-intercept

We require to find m and b to establish the equation.

To find 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)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (2 ,4) and (4 ,0)

let `(x_1,y_1)=(2,4)" and " (x_2,y_2)=(4,0)`

`rArrm=(0-4)/(4-2)=(-4)/2=-2`

We can write the partial equation as `y=-2x+b`

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

Using (4 ,0), that is x = 4 and y = 0

`rArr0=(-2xx4)+brArr0=-8+brArrb=8`

`rArry=-2x+8" is the equation"`

Answer 2

`2x+y=8`

Explanation:

If two coordinates are known a more direct formula is;

`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`

`(x_1,y_1)=(2,4)`

`(x_2,y_2)=(4,0)`

`(y-4)/(0-4)=(x-2)/(4-2`

`y/-4=(x-4)/2`

`2y=-4x+8`

`4x+2y=16`

`2x+y=8`