A wire is attached to the top of a pole. The wire is 7 ft longer than the pole, and the distance from the wire to the bottom of the pole is 7 ft less

Step 1
Let x be the height of the pole so that x+7 is the lenght of the wire and the distance from the wire to the bottom of the pole is x - 7, all in ft.
Label the right triangle as shown:

Step 2
Using Pythagorean Theorem, we solve for x:
${\displaystyle x^2+{(x-7)}^2={(x+7)}^2}$
${\displaystyle x^2+x^2-14x+49=x^2+14x+49}$
${\displaystyle 2x^2-14x+49=x^2+14x+49}$
${\displaystyle x^2-28x=0}$
Factor:
x(x-28)=0
By zero product property,
x=0,28
The height of the pole cannot be 0 so we tale x = 28 as the solution.
So, the lenght of the wire is 28 + 7 = 35 ft and the height of the pole is 28 ft.
Result:
The lenght of the wire is 35 ft, and the height of the pole is 28 ft.