CMR : 7/y = x-z / z-y

To prove the given equation, we'll start by simplifying the right-hand side of the equation:

1/2 * (1/x - 1/y)
= 1/2 * ((y - x)/(xy))
= (y - x)/(2xy)

Now, let's substitute this expression into the left-hand side of the equation:

1/7 = (y - x)/(2xy)

To get rid of the fractions, we'll multiply both sides of the equation by 7:

1 = 7(y - x)/(2xy)

Next, we'll cross-multiply:

2xy = 7(y - x)

Expanding the equation:

2xy = 7y - 7x

Now, let's rearrange the equation to isolate the variables:

7y - 2xy = 7x

Dividing both sides of the equation by 7:

y - (2x/7)y = x

Factoring out y on the left-hand side:

y(1 - 2x/7) = x

Dividing both sides of the equation by (1 - 2x/7):

y = x / (1 - 2x/7)

Now, let's substitute this expression for y into the right-hand side of the equation:

7/y = 7 / (x / (1 - 2x/7))

Simplifying the expression:

7/y = 7(1 - 2x/7) / x
= 7(7 - 2x) / 7x
= (7 - 2x) / x

Finally, we can see that:

(7 - 2x) / x = x - z / z - y

Therefore, we have proved that:

7/y = x - z / z - y