Question

If 3^x = 5^y = 225^z, find the relation between x, y and z.

Answer 1

Given:

3^x = 5^y = 225^z

Step-wise explanation:

1. Let the common value be k:

3x = k

5y = k

225z = k

2. From these, express x, y and z in terms of k:

`x = k/3`

`y = k/5`

`z = k/225`

3. The expression relates z, x and y as:

`2z = (xy)/(x + y)`

4. Substitute the values of x, y, z:

Left side: `2z = 2 xx (k/225) = (2k)/225`

Right side: `(xy)/(x + y) = ((k/3) xx (k/5))/((k/3) + (k/5))`

5. Simplify the right side:

`(k/3) xx (k/5) = k^2/15`

`(k/3) + (k/5) = (5k + 3k)/15`

`(k/3) + (k/5) = (8k)/15`

So, `(xy)/(x + y) = (k^2/15)/((8k)/15)`

`(xy)/(x + y) = (k^2/15) xx (15/(8k))`

`(xy)/(x + y) = k/8`

6. Equate left and right side:

`(2k)/225 = k/8`

Multiply both sides by 225 × 8:

2k × 8 = k × 225

16k = 225k

7. This equality holds only if k = 0 or the original relation is incorrect.

So, the direct substitution likely arises from another relation involving x, y and z.