Mathcounts Prep : Algebra Manipulation

Note that: 

${(x+y)}^2-2xy=x^2+y^2$
${(x-y)}^2+2xy=x^2+y^2$
${(x-y)}^3+3xy(x-y)=x^3-y^3$
${(x+y)}^3-3xy(x+y)=x^3+y^3$
${(x+y+z)}^2-2(xy+yz+xz)=x^2+y^2+z^2$

Applicable questions:

Question 1: If x + y = a and xy = b, what is the sum of the reciprocals of x and y?

Solution: 
${\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{y}}={\displaystyle \frac{x+y}{xy}}$= ${\displaystyle \frac{a}{b}}$

Question 2: If $x^2+y^2=153$ and x + y = 15, what is xy?

Solution: 
${(x+y)}^2-2xy=x^2+y^2$
${15}^2-2xy=153$$\to xy=36$

Question 3: If ${(x+y)}^2=1024$ , $x^2+y^2$ = 530 and x > y , what is x - y? 

Solution: 
 ${(x+y)}^2-2xy=x^2+y^2$
1024 - 2xy = 530$\to 2xy=494$
${(x-y)}^2+2xy=x^2+y^2$
${(x-y)}^2=36$
x - y = 6

Question 4: x + y = 3 and  $x^2+y^2=89$, what is $x^3+y^3$? 

Solution: 
${(x+y)}^2-2xy=x^2+y^2$
9 - 2xy = 89 $\to -2xy=80$ so xy = -40
${(x+y)}^3-3xy(x+y)=27-3(-40)\ast 3=27+3\ast 40\ast 3=x^3+y^3$
$x^3+y^3$= 387

Question #5: If $x+{\displaystyle \frac{1}{x}}=5$, what is $x^3+{\displaystyle \frac{1}{x^3}}$?

Solution:
${(x+{\displaystyle \frac{1}{x}})}^3=x^3+3x^2.{\displaystyle \frac{1}{x}}+3x.{\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{x^3}}$
$5^3=x^3+3(x+{\displaystyle \frac{1}{x}})+{\displaystyle \frac{1}{x^3}}$
125 - 3*5 = $x^3+{\displaystyle \frac{1}{x^3}}$
The answer is 110.

Question #6 : 2011 Mathcounts state sprint #24 : x + y + z = 7 and $x^2+y^2+z^2=19$, what is the arithmetic mean of the three product xy + yz + xz? 

Solution: 
${(x+y+z)}^2-2(xy+yz+xz)=x^2+y^2+z^2$
$7^2-2(xy+yz+xz)=19$
xy + yz + xz = 15  so their mean is ${\displaystyle \frac{15}{3}}=5$



More practice problems (answer key below):

#1:If x + y = 5 and xy = 3, find the value of ${\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{y^2}}$. 

#2: If x + y = 3 and $x^2+y^2=6$, what is the value of $x^3+y^3$? 

#3: The sum of two numbers is 2. The product of the same two numbers is 5. 
 Find the sum of the reciprocals of these two numbers, and express it in simplest form. 

#4:If $x-{\displaystyle \frac{6}{x}}$ = 11, find the value of $x^3-{\displaystyle \frac{216}{x^3}}$? 

#5: If $x+{\displaystyle \frac{3}{x}}=9$, find the value of $x^3+{\displaystyle \frac{27}{x^3}}$?

#6:If $x+{\displaystyle \frac{1}{x}}=8$, what is $x^3+{\displaystyle \frac{1}{x^3}}$? 





Answers:
#1 :${\displaystyle \frac{19}{9}}$
#2: 13.5
#3: ${\displaystyle \frac{2}{5}}$
#4: 1529 [ ${11}^3$+ 3 x 6 x 11 =1529]
#5: 648   [$9^3$-3 x 3 x 9 = 648]
#6: 488   [ $8^3$– 3 x 8 = 488]