Question

What is the equation of the line passing through `(34,5)` and `(4,-31)`?

Answer 1

`y = (6x-179)/5`.

Explanation:

We will set up the co-ordinates as:
`(34, 5)`
`(4, -31)`.

Now we do subtraction of the `x`s and the `y`s.

`34 - 4 = 30`,
`5 -(-31) = 36`.

We now divide the difference in `y` over that in `x`.

`36/30 = 6/5`.
So `m` (gradient) `= 6/5`.

Equation of a straight line:
`y = mx +c`. So, let's find `c`. We substitute values of any of the coordinates and of `m`:
`5 = 6/5 * 34 + c`,
`5 = 204/5 + c`,
`c = 5 - 204/5`,
`c = -179/5`. So,

`y = (6x-179)/5`.

Answer 2

`color(blue)(y= 6/5x-35.8)`

Explanation:

Standard form equation is:

`color(blue)(y=mx+c............................(1))`

Where m is the slope (gradient) and c is the point where the plot crosses the y-axis in this context.

The gradient is the amount of up (or down) of y for the amount of along for the x-axis. `color(blue)("Always considered from left to right .")`

So `m -> (y_2-y_1)/(x_2-x_1) = ((-31)-5)/(4-34)`

As `(34,5)` is listed first you assume this is the left most point of the two.

`m= (-36)/(-30)` dividing negative into negative gives positive

`color(blue)(m=(36)/(30) = 6/5 .........................(2))`

Substitute (2) into (1) giving:

`color(blue)(y=6/5x+c............................(3))`

Now all we need to do is substitute known values for x and y to obtain that for c

Let `(x,y) -> (34,5)`

Then `y=6/5x+c" "` becomes:

`color(brown)(5=(6/5 times 34) +c)` `color(white)(xxx)`brackets used for grouping only

Subtract `color(green)((6/5 times 34))` from both sides giving

`color(brown)(5) -color(green)((6/5 times 34)) color(white)(xx) = color(white)(xx)color(brown)( ( 6/5 times 34)) -color(green)((6/5 times 34))color(brown)( +c)`

`c=5-(6/5times 34)`

`color(blue)(c = -35.8....................................(4))`

Substitute (4) into (3) giving:

`color(blue)(y= 6/5x-35.8)`