Question 28529
The line through points (m,-9) and (7,m) has a slope of m. What is the value of m?
The slope of a line joining two given points A(x1,y1) and B(x2,y2) is given by slope =(y2-y1)/(x2-x1)    -----(1)
In the given problem we have A(X1,y1) given by A(m,-9) implying x1=m, y1=-9
and B(X2,y2) given by B(7,m) implying x2=7, y2=m
And given slope = m
Therefore using the formula (1), we have
m = [m-(-9)]/[7-m]
m(7-m) = (m+9)  [multiplying through out by (7-m)]
7m-m^2 = m +9
0 = m^2-7m+m+9   
(taking both the terms from the left to the right, change side then change sign)
That is 0 = m^2-6m+9
0= (m-3)^2   
[as (m^2-6m+9) is a perfect square of the form a^2-2ab+b^2 
where here a = m and b=3]
And (m-3)^2  = 0 implies m=3 (the equation has root 3 occurring twice)
Answer: m = 3
Verifiation: The points A(m,-9) and B(7,m) become 
A(3,-9) and B(7,3) 
and the slope of the line through A(3,-9) and B(7,3) by formula (1) is 
[3-(-9)]/(7-3) = (3+9)/4 = 12/4 = 3 = m which is the given slope.
Hence our answer for m is right