Question

How do you write an equation of a line given slope is 3/7 and the ordered pair is (0, 5/6)?

Answer 1

`y=3/7x+5/6`

Explanation:

`color(red)("Step 1")`
The general equation of a straight line is: `y=mx+c`

Where `m` is the gradient and `c` is a constant which is also the y-intercept.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)("Step 2")`

We are told that the gradient (slope) is `3/7` so by substituting this for `m` we have

`y=3/7x+c`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(red)("Step 3")`

We are given a condition where this equation is true. We are told that it passes through the point `(x,y)->(color(blue)(0,5/6))`

So we van substitute for both `x` and `y` to determine the value of `c`.

`color(brown)(y=3/7x+x" "->" "color(blue)(5/6)=3/7(color(blue)(0))+c)`

But `7/7xx0=0` giving:

`5/6=c`

Thus the finished equation is:

`y=3/7x+5/6`

Answer 2

`y = 3/7x +5/6`

Explanation:

There is a nifty formula for the equation of a line which applies in just such a case where we are given the slope and one point.

Using this formula requires only ONE step with substitution and some easy simplifying. Works like a dream...

`y-y_1 = m(x-x_1)" m = slope and "(x_1,y_1)" is a point"`

`y -5/6 = 3/7(x-0)`

`y = 3/7x +5/6`

However, in this particular case, no working was required at all.

The slope, m is given as `3/7` and the point given happens to be the y-intercept because the `x-`value is 0. So `c = 5/6`

`y = mx + c rArr y = 3/7x +5/6`

Answer 3

Alternative approach

`y=3/7x+5/6`

Explanation:

Let the slope (gradient) be `m`

The gradient (slope) the change in up or down for a given change in along.

`m=("Change in y")/("Change in x") = 3/7`

Let given point be `P_o->(x_o,y_o) = (0,5/6)`

Let any other point be `P_i->(x_i,y_i)`

Then `m=("Change in y")/("Change in x") -> (y_i-y_o)/(x_i-x_o) = (y_i-5/6)/(x_i-0) = 3/7`

`y_i=3/7x_i+5/6`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`y=3/7x+5/6`