Systems of Equations

System of Equations - Problem Solving


OABOAB is a triangle in the first quadrant of the xy-plane with vertices O(0,0)O(0,0), A(5,0)A(5,0) and B(x,y)B(x,y), where xx and yy are positive real numbers. If OB=6OB = 6 and AB=3AB = 3, then y2y^2 can be written as ab\frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

What is a4+b4+c4a^4 + b^4 + c^4?

In a school, there are 44 more students in Class A than in Class B, and 44 more students in Class B than in Class C. As a reward for good behavior, a teacher decides to give out some sweets. Every student in class C got 55 more sweets than every student in Class B, and every student in Class B got 33 more sweets than every student in Class A. As it turns out, Class A got 1010 more sweets than Class B, and Class B got 1414 more sweets than Class C. How many sweets were given out in all?

Given the system of equations {x(x+y)=9y(x+y)=16 \begin{cases} x(x+y) &=& 9 \\ y(x+y) &=& 16 \end{cases} the value of xyxy can be written as ab\frac{a}{b} where aa and bb are positive coprime integers. Find a+ba+b.

An ordered triple of real numbers (a,b,c) (a, b, c) is called friendly, if each number is equal to the product of the other 2. How many (distinct) friendly triples are there?

Details and assumptions

The numbers need not be pairwise distinct (which means that no two of them are the same).


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