Homework Help

solve for x and y -- x^2 + y^2 = 25 x^3 + y^3 = 91

user profile pic

rnk | Student, Grade 10 | Honors

Posted September 4, 2010 at 4:06 PM via web

dislike 0 like

solve for x and y --

x^2 + y^2 = 25

x^3 + y^3 = 91

Tagged with indices, math

3 Answers | Add Yours

Top Answer

user profile pic

neela | High School Teacher | Valedictorian

Posted September 4, 2010 at 9:57 PM (Answer #1)

dislike 2 like

x^2+y^2 = 25..(1)

x^3+y^3 = 91..(2)

To solve for x and y.

Solution:

By guess the solution  x = 4 and y = 3. Or y = 3 and x = 4, as

x^2+y^2 = 4^2+3^2 = 25 and

x^3+y^3 = 4^3+3^3 = 64+27 = 91.

We knowthat x^3+y^3 = (x+y) (x^2+y^2 -xy).

Therefore,  (x+y)(x^2+y^2-xy) =  91 for the given data.

 (x+y)(x^2+y^2-xy) = 13*7.  Or 7*13.

Let us check for x+y = 7 and x^2+y^2-xy  = (x+y)^2-3xy = 7^2 -3xy = 13 . Or

49-3xy = 13.

3xy = 49-13 = 36

xy = 36/4 =12.

So x+y =7..........(1). Therefore,

x-y = sqrt{(x+y)^2 -4xy  }

x-y = {49-48} = 1.

x-y = 1...............(2).

Add eq(1) and eq(2):

2x = 7+1=8

x= 8/2 = 4.

Eq(1)-eq(2) gives:

 2y = 7-1 = 6.

y = 6/2 = 3.

Therefore  (x , y) = (4,3). Also (x,y) = (3,4).

 

 

user profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted September 4, 2010 at 7:22 PM (Answer #2)

dislike 0 like

x^2 + y^2 = 25........(1)

x^3 + y^3 = 91 .......(2)

We know that:

3^2 + 4^2 = 5^2

==> x = 3

==> y= 4

To check the answer, substitute in (2):

x^3 + y^3 = 91

3^3 + 4^3 = 91

27 + 64 = 91

91 = 91

Then the answer is:

x= 3    andy  y = 4   OR

x= 4    and    y = 3

user profile pic

giorgiana1976 | College Teacher | Valedictorian

Posted September 5, 2010 at 11:51 PM (Answer #3)

dislike -1 like

We'll solve the system, starting from the following formulas:

x^2 + y^2  = (x+y)^2 - 2xy

x^3 + y^3 = (x+y)^3 - 3xy(x+y)

Now, we'll note x+y = S and xy = P.

x^2 + y^2  = S^2 - 2P

x^3 + y^3 = S^3 - 3SP

We'll substitute them into the equations of the system:

S^2 - 2P = 25 (1)

S^3 - 3SP = 91 (2)

We'll factorize (2):

S(S^2 - 3P) = 91

We'll extract S^2 from (1)=>S^2 = 25+2P

S(25+2P - 3P) = 91

S(25-P) = 91

We'll remove the brackets:

25S - SP = 91 => SP = 25S - 91

S^3 - 3SP = 91

S^3 - 3 (25S - 91) = 91

S^3 - 75S = 91 - 3*91

S(S^2 - 75) = -2*91

S = -2

S^2 - 75 = 91

S^2 = 91+75

S^2 = 166

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes