Assume P(xp,yp), A(xa,ya), etc.
We know that rotation rule of 90<span>° clockwise about the origin is
R_-90(x,y) -> (y,-x)
For example, rotating A about the origin 90</span><span>° clockwise is
(xa,ya) -> (ya, -xa)
or for a point at H(5,2), after rotation, H'(2,-5), etc.
To rotate about P, we need to translate the point to the origin, rotate, then translate back. The rule for translation is
T_(dx,dy) (x,y) -> (x+dx, y+dy)
So with the translation set at the coordinates of P, and combining the rotation with the translations, the complete rule is:
T_(xp,yp) R_(-90) T_(-xp,-yp) (x,y)
-> </span>T_(xp,yp) R_(-90) (x-xp, y-yp)
-> T_(xp,yp) (y-yp, -(x-xp))
-> (y-yp+xp, -x+xp+yp)
Example: rotate point A(7,3) about point P(4,2)
=> x=7, y=3, xp=4, yp=2
=> A'(3-2+4, -7+4+2) => A'(5,-1)
Answer:
We cannot say that the mean wake time are different before and after the treatment, with 98% certainty. So the zopiclone doesn't appear to be effective.
Step-by-step explanation:
The goal of this analysis is to determine if the mean wake time before the treatment is statistically significant. The question informed us the mean wake time before and after the treatment, the number of subjects and the standard deviation of the sample after treatment. So using the formula, we can calculate the confidence interval as following:
![IC[\mu ; 98\%] = \overline{y} \pm t_{0.99,n-1}\sqrt{\frac{Var(y)}{n}}](https://tex.z-dn.net/?f=IC%5B%5Cmu%20%3B%2098%5C%25%5D%20%3D%20%5Coverline%7By%7D%20%5Cpm%20t_%7B0.99%2Cn-1%7D%5Csqrt%7B%5Cfrac%7BVar%28y%29%7D%7Bn%7D%7D)
Knowing that 
:
![IC[\mu ; 98\%] = 98.9 \pm 2.602\frac{42.3}{4} \Rightarrow 98.9 \pm 27.516](https://tex.z-dn.net/?f=IC%5B%5Cmu%20%3B%2098%5C%25%5D%20%3D%2098.9%20%5Cpm%202.602%5Cfrac%7B42.3%7D%7B4%7D%20%5CRightarrow%2098.9%20%5Cpm%2027.516)
![IC[\mu ; 98\%] = [71.387 ; 126,416]](https://tex.z-dn.net/?f=IC%5B%5Cmu%20%3B%2098%5C%25%5D%20%3D%20%5B71.387%20%3B%20126%2C416%5D)
Note that ![102.8 \in [71.384 ; 126.416]](https://tex.z-dn.net/?f=102.8%20%5Cin%20%5B71.384%20%3B%20126.416%5D)
so we cannot say, with 98% confidence, that the mean wake time before treatment is different than the mean wake time after treatment. So the zopiclone doesn't appear to be effective.
3x-5y=-3 original problem
3(1)-5y=-3 put 1 in the place of x and multiply
3-5y=-3 subtract 3 from both sides
-5y=-6 divide both sides by -5
y=possitive 6/5
hope that helps
