suppose x and y are inversely proportional.
a)if x^p and y^q are also inversely proportional, then how must p and q be related?
b)if x^p and y^q are directly proportional, then how must p and q be related?
a) i find if you add 1 to p (multiply by x) you add 1 to q
but to keep the thing constant you must divide it by xy so you dont change anything? i dont understand
To understand the relationship between powers p and q when x^p and y^q are inversely proportional, let's break it down step by step.
When two variables are inversely proportional, it means that as one variable increases, the other variable decreases in such a way that their product remains constant. In mathematical terms, it can be represented as:
xy = k
where k is a constant.
Now, let's consider the situation where x^p and y^q are also inversely proportional.
From the given information, we can write:
(x^p)(y^q) = k
To simplify this expression, we can use the property of exponents that states (a^m)(a^n) = a^(m+n).
Using this property, we get:
x^p * y^q = x^p+y^q = k
Comparing this equation with the previous one, we can see that the powers p and q are simply added together. Therefore, we can conclude that when x^p and y^q are inversely proportional, the powers p and q are related by the equation:
p + q = c
where c is a constant.
Now, let's move on to the second part of your question.
b) If x^p and y^q are directly proportional, it means that as one variable increases, the other variable also increases in such a way that their ratio remains constant. In mathematical terms, we can represent it as:
(x^p) / (y^q) = k
where k is a constant.
To find the relationship between the powers p and q in this situation, we can simplify the expression by using the quotient rule of exponents that states (a^m) / (a^n) = a^(m-n).
Applying this rule to the given expression, we have:
(x^p) / (y^q) = x^(p-q) = k
Comparing this equation with the previous one, we can see that the power p is subtracted by the power q. Therefore, we can conclude that when x^p and y^q are directly proportional, the powers p and q are related by the equation:
p - q = c
where c is a constant.