#algebra #matholympiad a+b=abc, b+c=abc, c+a=abc find a, b, c
#matholympiad #olympiad a+b=abc, b+c=abc, c+a=abc find a, b, c
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Video Summary
AI GeneratedThis video provides a step-by-step mathematical derivation to solve a system of three equations from a Math Olympiad problem: $a + b = abc$, $b + c = abc$, and $c + a = abc$. The instructor demonstrates how to prove that $a = b = c$ and calculates the three possible sets of values for the variables.
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Please the way like that in this video. We are going to discuss the problem of math Olympiad
And the problem is a plus B equal to ABC B plus C equal to ABC
C plus a equal to ABC find the value of a B and C
So we have given three equation a plus B equal to ABC
say equation number 1 B plus C equal to ABC equation number 2 and C plus a equal to ABC equation number 3 in
The first step we are going to take the subtraction of 1 and 2
Then we have a plus B equal to ABC minus B plus C equal to ABC. So here
Plus B cancel with the minus B
Plus ABC cancel with the minus ABC and we have a minus C equal to 0
So a equal to C say equation number 4 in the second step
We take the subtraction of 2 and 3 then we have B plus C equal to ABC
minus C plus a equal to ABC C cancel with the minus C and
Plus ABC cancel with the minus ABC then we have B minus a equal to 0
Therefore B equal to a say equation number 5
So by combining equation 4 and 5 we have B equal to a and a equal to C
Therefore we have a equal to B equal to C say equation number 6
Therefore we have 4 equation
first 3 equation given in the data and the last equation we derive a equal to B equal to C now substitute the
value of 6 equation a equal to B equal to C in the first equation a
Plus B equal to ABC here B equal to a and C equal to it
So therefore we have a plus a equal to a into a into so it will give 2 a equal to a cube
We can rewrite a cube minus 2 a equal to 0
Therefore a is a common factor. So a into a square minus 2 equal to 0
So it will give 2 equation a equal to 0 and a square minus 2 equal to 0
Therefore a square equal to 2
So a has 2 possibility a equal to plus or minus square root 2
So a equal to 0 a equal to square root 2 and a equal to minus square root 2
since
Equation number 6 is that a equal to B equal to C. So possible values for ABC are if
A equal to 0 then B and C are also 0 the first possibility is 0 0 0
If a equal to square root 2 then B and C also square root 2
So therefore second possibility is square root 2 square root 2 square root 2 and a equal to minus square root 2
Then third possibility is minus square root 2 minus square root 2 minus square root 2
So from this 3 equation, we have 3 possibility of ABC
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