(1) The family of linear equations with slope 4 is `f(x)=4x+k`

** We have `f'(x)=4` , so `int 4dx=4x+C` , replace the constant of integration with parameter `k` **

(2) The family of linear equations with `f(4)=1` is `g(x)=a(x-4)+1`

(3) The equation(s) that are in both families satisfy both conditions.

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(3) The equation(s) that are in both families satisfy both conditions.

Then:

`4x+k=a(x-4)+1`

`4x+k=ax-4a+1`

`(4-a)x+(k+4a-1)=0`

When x=4 we have:

`(4-a)4+k+4a-1=0`

`16-4a+k+4a-1=0`

`k=-15`

**Thus the equation that lies in both families is `h(x)=4x-15` **

Linear functions can be expressed as

`f(x)=mx+b`

So an example of a linear function with slope 4 is

`f(x)=4x+7`

Now let's make up a random linear equation such that f(4) = 1.

How about `f(x)=0x+1` ? That's a horizontal line through (4,1). Or maybe `f(x) = x-3` ? This would also pass through the point (4,1).

But if the slope is 4 *and* the function passes through (4,1), we have

`f(4) = 4*4+b=1`

so b, the y-intercept, is -15.

`f(x) = 4x - 15`