Is rise over run the way you find slope?

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Rise over run is indeed the method used to find the slope of a line, represented mathematically as m = (Y2 - Y1) / (X2 - X1). This formula calculates the change in y-values (rise) divided by the change in x-values (run). The slope can also be expressed as m = Δy / Δx, emphasizing the concept of change in both dimensions. Understanding this relationship is fundamental in algebra and geometry. Therefore, rise over run is a key principle in determining slope.
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Is rise over run the way you find slope?
 
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Yes it is. Its like saying

m = \frac{Y_2-Y_1}{X_2-X_1}
 
Indeed, note that it is the 'change' in y-values divided by the 'change' in x-values. We can write this as (and this is equivalent to the expression above):

m = \frac{{\Delta y}}{{\Delta x}}
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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