Question

How do you show that the slope of `{(y=at+b),(x=ct+d):}` is `a/c`?

Answer 1

Please see the explanation below

Explanation:

Each equation is defined using the independent variable `t`

As `t` changes `x` and `y` change accordingly.

Both `x` and `y` are linear functions

For `y` it is a line with slope `a`

We can write this as follows `(\Delta y)/(\Delta t)=a`

Solving for `\Delta y`

`\Delta y=a\Deltat`

For `x` it is a line with slope `c`

Proceeding as before we have

`(\Deltax)/(\Deltat)=c`

Solving for `\Deltax`

`\Deltax=c\Deltat`

The slope of the overall function is

`(\Deltay)/(\Deltax)`

Therefore

`(\Deltay)/(\Deltax)=(a\Deltat)/(c\Deltat)=a/c`

NOTE: `\Delta` means "change in"