two quantities M and n ARE SUCH THAT m is directly prportional to x. When x=2, their sum is 8 and when x=m their sum is 7. Determine the re;ationship between m and n
Let's assume that the relationship between M and n is given by M = k * n, where k is the constant of proportionality.
According to the problem statement, M is directly proportional to x. This implies that when x doubles, M also doubles.
Therefore, when x = 2, M = 2 * k * n = 2k * n.
We are given that the sum of M and n is 8 when x = 2, so we can write the equation:
2k * n + n = 8
Similarly, when x = M, the sum of M and n is 7. Using our assumption, we can write the equation:
k * n + n = 7
Simplifying both equations, we get:
(2k + 1) * n = 8
(k + 1) * n = 7
Since the left sides of both equations are equal to n, we can equate the right sides:
2k + 1 = k + 1
Solving for k, we find:
k = 0
Now that we have the value of k, we can determine the relationship between M and n:
M = k * n = 0 * n = 0
Therefore, the relationship between M and n is M = 0.
To determine the relationship between the quantities M and n, we need to use the given information about the proportionality between M and x.
First, let's write the statement of proportionality:
M ∝ x
Next, we can introduce a constant of proportionality, k, such that:
M = kx
Now, let's analyze the given information:
1. When x = 2, the sum of M and n is 8.
We can substitute x = 2 into the equation M = kx to get:
M = 2k
M + n = 8
2. When x = M, the sum of M and n is 7.
We can substitute M for x in the equation M = kx to get:
M = Mk
M + n = 7
From these two equations, we have a system of equations:
1) M + n = 8
2) M + n = 7
It is worth noting that the equations contradict each other because they both state different sums while x=M. Therefore, it seems that there is a mistake or inconsistency in the given information. Please double-check the problem statement to ensure accuracy.
If you have any other questions, feel free to ask!
To determine the relationship between M and n, we need to analyze the given information step-by-step.
Step 1: Understanding the problem.
The problem states that two quantities, M and n, are related to x. It is mentioned that M is directly proportional to x.
Step 2: Understanding direct proportionality.
When two quantities are directly proportional, it means that their ratio remains constant. Mathematically, if M is directly proportional to x, we can write it as:
M ∝ x
Step 3: Expressing the proportionality as an equation.
When we have a direct proportion, we can introduce a constant of proportionality (k) to express the relationship. By doing so, we get the equation:
M = kx
Step 4: Using the given information to find the constant of proportionality.
We are given two data points:
When x = 2, the sum of M and n is 8: M + n = 8
When x = M, the sum of M and n is 7: M + n = 7
Step 5: Solving for the constant of proportionality.
Using the equation M = kx, we can solve for the constant of proportionality (k) in the given data points.
When x = 2, we have:
M = k(2) = 2k
When x = M, we have:
M = k(M) = kM
From these equations, we can obtain two equations:
2k + n = 8 (equation 1)
kM + n = 7 (equation 2)
Step 6: Solve the system of equations.
To find the relationship between M and n, we need to solve the system of equations formed in step 5.
Subtracting equation 2 from equation 1, we get:
(2k + n) - (kM + n) = 8 - 7
2k - kM = 1
Factoring out the k:
k(2 - M) = 1
Dividing both sides by (2 - M):
k = 1 / (2 - M)
Therefore, the relationship between M and n is:
n = 8 - 2k
Substituting the value of k from step 6:
n = 8 - 2(1 / (2 - M))
Simplifying the equation:
n = 8 - 2 / (2 - M)
So, the relationship between M and n is:
n = 8 - 2 / (2 - M)