Y varies partly as x, partly inversely as x and partly constant. The graph of y against x passes through the points (2,20), (3,16), and (5,14). Find a general formula for y in terms of x
My proposed formula is
y = x / kx
Your equation just satisfies y varies directly as x. What about the rest?
y = ax + b/x + c
Now use the points to get 3 equations in a,b,c and solve them.
To find a general formula for y in terms of x, let's first analyze the given information.
We are given that y varies partly as x, partly inversely as x, and partly constant.
When y varies partly as x, we can represent this relationship with the equation y = kx, where k is a constant.
When y varies inversely as x, we can represent this relationship as y = k/x, where k is another constant.
Finally, when y is constant, regardless of the value of x, we can represent this relationship as y = k, where k is yet another constant.
Now, let's use the given points (2,20), (3,16), and (5,14) to solve for the values of k and determine the appropriate equations depending on the relationship stated.
Using the first point (2,20):
20 = k * 2
Simplifying, we find:
k = 10
Using the second point (3,16):
16 = k * (1/3)
Simplifying, we find:
k = 48
However, this value of k contradicts the value we obtained from the first point. Therefore, the relationship y = k * (1/x) is not applicable in this case.
We can conclude that the relationship for y must be a combination of y = kx and y = k.
Using the third point (5,14):
14 = 5k
We find:
k = 14/5 = 2.8
Therefore, the general formula for y in terms of x is:
y = 2.8x
Here, the value of k is 2.8, which is obtained from the constant relationship in the third point.
To find a general formula for y in terms of x, we can use the information given in the problem.
First, let's consider the part where y varies partly as x. If y varies directly with x, it means that y is proportional to x. We can write this as y = kx, where k is the proportionality constant.
Next, let's consider the part where y varies partly inversely as x. If y varies inversely with x, it means that the product of y and x is a constant. We can write this as xy = c, where c is a constant.
Finally, we have the part where y is constant. This means that y does not depend on x and remains constant regardless of the value of x.
Now, by using the given points (2,20), (3,16), and (5,14), we can substitute these values into the equations and solve for the constants.
For the point (2,20):
Using y = kx, we have 20 = k * 2, which gives us k = 10.
For the point (3,16):
Using xy = c, we have 3 * 16 = c, which gives us c = 48.
Since y is constant, the formula for this part is simply y = c, which in this case is y = 48.
Now, we have the following parts:
y = kx
xy = c
y = 48
To find the general formula for y in terms of x, we need to combine all these parts.
Taking the equation y = kx, we substitute the value we found for k:
y = 10x
Taking the equation xy = c, we solve for y:
y = c/x
Substituting the value we found for c, we have:
y = 48/x
Finally, combining all the parts, the general formula for y in terms of x is:
y = 10x + 48/x