I know that were doing problems were there is a soultion given which is equal to X^? times a constant ...
y = k x^?
So this is problem I have to decide given a probelm x relationship to y (for example iverse square cube and so forth) this unkown variable is denoted witha (?)
So given two points (Q,W) (E,R) how do I determiine the relationsip of x to y... and solve for the (?)
I guess so far in my book the constant is raised to the first power is there any way I could find out if the constant is not raised to first power or not???
Once I come up with the formula I know how to do the rest I just don't know how to obtain the formula and solve for the (?) once you do it's just plug and chug...
so please show me how to solve for (?) given two points
y = kx?
Substitute the x and y value, then take the log of both sides.
log y = k*?*log x
can you show me how to do this with the points
(2, 5)
(6, 80)
Two points determine a line. Given two points (Q,W) and (E,R), you can find the equation of the unique line, y = m x + b that goes through those two points.
Three points determine a parabola, y = a x^2 + b x + c
Four points determine a cubic, y = a x^3 + b x^2 + c x + d
etc
Now
given (Q,W) and (E,R)
slope = m = (R-W)/(E-Q)
so now you have m
go back and put first point in to get b from b = y - m x = W - m Q
ok
m = (80-5) / (6-2)
m = 75/4
then
b = 5 - (75/4)2
b = 5 - 75/2
b = (10-75)/2
b = -65/2
so
y = (75/4)x -65/2
or
4 y = 75 x - 130
I did not understand that your question related only to direct variation until I saw Dr. Bob's answer.
To determine the relationship between x and y and solve for the unknown exponent represented by the question mark (?, let's go through the process step by step:
1. Start with the general equation for the relationship between x and y: y = kx^?
2. Substitute the coordinates of the two given points into the equation. Let's assume the points are (Q, W) and (E, R). Replace x with Q and y with W in the first equation, and then replace x with E and y with R in the same equation. This will give you two equations to work with.
Equation 1: W = kQ^?
Equation 2: R = kE^?
3. Divide Equation 2 by Equation 1 to eliminate the constant k:
(R / W) = (kE^?) / (kQ^?)
4. Simplify the division:
(R / W) = (E^?) / (Q^?)
5. If the constant k is not raised to the first power, you would see it cancel out during the division and the resulting equation would not have it on either side.
6. Take the logarithm of both sides of the equation to help solve for the unknown exponent ?. The choice of logarithm base depends on what you are comfortable working with or what your book suggests. Natural logarithm (ln) or logarithm base 10 (log) are commonly used.
For example, if we use the natural logarithm:
ln(R / W) = ln((E^?) / (Q^?))
7. Apply the logarithm properties to simplify the equation further. One useful property is the logarithm of a fraction, which becomes the logarithm of the numerator minus the logarithm of the denominator.
ln(R / W) = ln(E^?) - ln(Q^?)
8. Another logarithm property allows you to bring the exponent down as a coefficient on the logarithm:
ln(R / W) = ? * ln(E) - ? * ln(Q)
9. Now, you have an equation in the form y = mx + b, where y is ln(R / W), x is ln(E) and the unknown exponent ? is the slope. To solve for ?, you can use the slope formula:
? = (ln(R / W) - ln(Q)) / (ln(E) - ln(Q))
10. Once you calculate the value of ?, you can substitute it back into one of the original equations (either Equation 1 or Equation 2) to find the value of the constant k. Then you will have the complete equation for the relationship between x and y.
Remember to be cautious when dealing with logarithms and check if the values you plug in are positive and non-zero if using a logarithm with a base other than e.