If y varies directly as x, and we know that y is 180 when x is n, while y also equals n when x is 5, what is the value of n?

To solve the problem, we need to understand the concept of direct variation. When we say that y varies directly as x, we can express this relationship mathematically as:

y = kx

where k is a constant of variation.

From the problem, we have the following two scenarios:

• 1. When x = n, y = 180:

Plugging these values into our direct variation formula:

180 = kn

From this, we can express k as:

k = 180/n

Next, we take a look at the second scenario:

• 2. When x = 5, y = n:

Again, substituting into the direct variation equation gives us:

n = k * 5

Substituting the expression we found for k into this equation:

n = (180/n) * 5

Now we have:

n = 900/n

To eliminate n from the denominator, we can multiply both sides by n:

n² = 900

Taking the square root of both sides, we find:

n = 30

Thus, the value of n is:

30

In summary, both conditions agree, confirming that when n = 30, the variations hold true.