Question

What is the equation of the line with slope `m= -7/3` that passes through `(-17/15,-5/24)`?

Answer 1

`y=-7/3x-977/120`

or

`7x+3y=-977/40`

or

`280x+120y=-977`

Explanation:

We are finding a line, so it needs to follow the linear form. The easiest way to find the equation in this instance is using the gradient-intercept formula. This is:

`y=mx+c`

Where `m` is the gradient and `c` is the `y`-intercept.

We already know what `m` is, so we can substitute it into the equation:

`m=-7/3`

`=>y=-7/3x+c`

So now we need to find c. To do this, we can sub in the values of the point we have `(-17/15, -5/24)` and solve for `c`.

`x=-17/15`

`y=-5/24`

`=>y=-7/3x+c`

Substitute the values in:
`=>-5/24=-7/3(-17/15)+c`

Apply the multiplication
`=>-5/24=(-7*-17)/(3*5)+c`

`=>-5/24=119/15+c`

Isolate the unknown constant, so bring all the numbers to one side of the by subtracting `-119/15`

`=>-5/24-119/15=cancel(119/15)+c-cancel(119/15)`

`=>-5/24-119/15=c`

Multiply numerator and denominator by a number to get a common denominator in both fractions to apply the subtraction

`=>(-5*5)/(24*5)-(119*8)/(15*8)=c`

`=>-25/120-952/120=c`

`=>(-25-952)/120=c`

`=>-977/120=c`

So now we can also substitute c into the equation:

`y=-7/3x+c`

`=>y=-7/3x-977/120`

We can also put this into the general form, which looks like:

`ax+by=c`

To do this, we can rearrange the gradient intercept formula into the general formula using the steps shown below:

`=>y=-7/3x-977/120`

We need to get rid of all the fractions first. So we multiply everything with a denominator (using the smaller one will make it easier in my opinion), and it should get rid of the fractions:

`=>3(y)=3(-7/3x-977/120)`

`=>3y=3*-7/3x-3*977/120`

`=>3y=(cancel(3)*-7)/cancel(3)x-(3*977)/120`

`=>3y=-7x-2931/120`

`=>3y=-7x-977/40`

Then bring the `x` value to the other side by adding `-7x` to both sides

`=>3y+7x=cancel(-7x)-977/40+cancel(7x)`

`=>7x+3y=-977/40`

If you want you can get rid of the fraction by multiplying both sides by 40:

`=>40(7x+3y)=40(-977/40)`

`=>40*7x+40*3y=(cancel(40)-977)/cancel(40)`

`=>280x+120y=-977`