Two variables m and n are such that m is directly proportional to x and n is inversely proportional to x.When x=2,their sum is 8 and when x=3,their sum is 7.Determine the relationship between m and n
sorry - wrong yet again.
m and n do not necessarily use the same value of k. So let's say
m = ax and n = b/x
when x=2, the sum is 2a + b/2 = 8
when x = 3, the sum is 3a + b/3 = 7
solving those two equations, we get
(a,b) = (1,12)
check:
2 + 12/2 = 8
3 + 12/3 = 7
Apologies for the incorrect previous response.
The relationship between m and n is as follows:
m = x
n = 12/x
This is because when x = 2, m = 2 and n = 6. When x = 3, m = 3 and n = 4.5.
To determine the relationship between m and n, we need to express m and n in terms of x.
Let's start with m. We know that m is directly proportional to x. This means that m can be expressed as:
m = kx
where k is the constant of proportionality.
Now let's move on to n. We know that n is inversely proportional to x. This means that n can be expressed as:
n = k/x
Note that the constant of proportionality is the same for both m and n since they are related to x in the same equation.
We are given that when x = 2, the sum of m and n is 8. Substituting the expressions for m and n into this equation, we get:
k(2) + k/(2) = 8
2k + k/2 = 8
To simplify the equation, we can multiply both sides by 2 to remove the fraction:
4k + k = 16
Combining like terms, we get:
5k = 16
Now, we can solve for k by dividing both sides by 5:
k = 16/5
So, the constant of proportionality, k, is 16/5.
Now that we have the value of k, we can express m and n in terms of x:
m = (16/5)x
n = (16/5)/x
In summary, the relationship between m and n is:
m = (16/5)x
n = (16/5)/x
If m is directly proportional to x, we can write this relationship as m = kx, where k is a constant of proportionality.
Similarly, if n is inversely proportional to x, we can write this relationship as n = k/x.
Now, we are given that when x = 2, the sum of m and n is 8. So we can write this as m + n = 8.
Substituting the expressions for m and n, we get k(2) + k/2 = 8.
Simplifying this equation, we have 2k + k/2 = 8. Multiplying through by 2 to remove the fraction, we get 4k + k = 16. Combining like terms, we have 5k = 16. Solving for k, we get k = 16/5.
Next, we are given that when x = 3, the sum of m and n is 7. So we can write this as m + n = 7.
Substituting the expressions for m and n, we get k(3) + k/3 = 7.
Simplifying this equation, we have 3k + k/3 = 7. Multiplying through by 3 to remove the fraction, we get 9k + k = 21. Combining like terms, we have 10k = 21. Solving for k, we get k = 21/10.
Since we have found two different values for k, we can conclude that there is no fixed relationship between m and n.