Question

What line is perpendicular to the line whose equation is `5y + 6 = -3x`?

Answer 1

Any line of form `y=5/3x+c`, where `c in RR` can be any real number.

Explanation:

The line `5y+6=-3x` has standard form `y=-3/5x-6/5`.

Hence its gradient (or slope) is `-3/5`.

From the theory we know that perpendicular lines have gradients whose product is `-1`.

So any line with gradient `5/3` will be perpendicular to the original given line. There are infinitely many such possibilities as you may vary the y-intercept as you please.

Answer 2

`y= 5/3x+c`

Explanation:

Given: `5y+6=-3x`

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

Standard equation form for a straight line graph is:
`y= mx +c` where m is the gradient.
`color(blue)("The gradient of a normal to this 'curve' is "(-1/m))`

Yes, technically you can call a straight line graph a curve! It is just that the gradient is constant.

As the gradient of the given curve `-> m -> -3/5`

Then the gradient of a curve that is 'normal' to it is:
`-1 times 1/m = (-1) times (-5/3) = (+ 5/3)`

No points were defined that this 'normal' should pass through so its intercepts are indeterminate. Hence you have to use the generic equation but using the new gradient. So we have:

`y= 5/3x+c`