y varies partly as x, partly inversely as x and partly constant. the graph of y againts x through the points (2.20), (3.16), and (5,14). Find a general formula for y in terms of x
y = ax + b/x + c
where a, b, and c are constants.
to find the particular values of a,b,c use
2a + b/2 + c = 20
3a + b/3 + c = 16
5a + b/5 + c = 14
solve those to get
y = x + 30/x + 3
Now finishing the question: plug in the 3 points
20 = 2a + b/2 + c or 40 = 4a + b + 2c
16 = 3a + b/3 + c or 48 = 9a + b + 3c
14 = 5a + b/5 + c or 70 = 25a + b + 5c
subtract the 1st from the 2nd -> 5a + c = -4
subtract the 2nd from the 3rd -> 16a + 2c = -2
double 5a + c = -4 to get 10a + 2c = -8
subtract from 16a + 2c = -2
6a = 6
a = 1
back in 5a + c = -4
c = -9
back into the first:
4a + b + 2c = 20
4 + b - 18 = 20
b = 34
y = x + 34/x - 9
To find a general formula for y in terms of x, we need to understand the relationship between y and x based on the given information.
The problem states that y varies partly as x, partly inversely as x, and partly constant. This means that the relationship between y and x can be expressed as a combination of direct variation, inverse variation, and constant terms.
Let's break down the problem step by step:
1. Direct Variation: The term "y varies partly as x" suggests that there is a direct variation between y and x. In a direct variation, y is directly proportional to x and can be expressed as y = kx, where k is the constant of variation.
2. Inverse Variation: The term "y varies partly inversely as x" implies that there is an inverse variation between y and x. In an inverse variation, y is inversely proportional to x and can be expressed as y = k/x, where k is the constant of variation.
3. Constant Term: The problem also mentions a constant term, which means there is a fixed value added to the relationship between y and x. Let's call this constant term c.
Now, we can write the general formula for y in terms of x by combining these three components:
y = kx + c + k/x
To find the specific values of k and c, we will use the given data points (2, 20), (3, 16), and (5, 14).
Substituting the values from the first data point (2, 20) into the general formula:
20 = 2k + c + k/2
Simplifying this equation gives: 20 = (4k + 2c + k) / 2
Multiplying both sides by 2 yields: 40 = 4k + 2c + k
Simplifying further: 40 = 5k + 2c
Using the second data point (3, 16):
16 = 3k + c + k/3
Multiplying both sides by 3 gives: 48 = 9k + 3c + k
Simplifying further: 48 = 10k + 3c
Using the third data point (5, 14):
14 = 5k + c + k/5
Multiplying both sides by 5 gives: 70 = 25k + 5c + k
Simplifying further: 70 = 26k + 5c
Now, we have a system of three equations:
40 = 5k + 2c
48 = 10k + 3c
70 = 26k + 5c
We can solve this system of equations to find the values of k and c. After finding k and c, you can substitute them back into the general formula for y in terms of x: y = kx + c + k/x, and you will have the specific formula for y in terms of x based on the given data points.