You are here:

# Algebra/equation

Question
Solve for x, y and z if: x + y + z = 1 ; x^2 + y^2 +z^2 = 35 and x^3 + y^3 + z^3 = 97. Thanks.

x + y + z = 1

(x + y + z)^2 = 1

x^2 + y^2 +z^2 + 2xy + 2xz + 2yz = 1

35 + 2xy +2xz + 2yz = 1

2xy +2xz + 2yz = -34

xy + xz + yz = -17

-17 = (1)(xy + xz + yz ) = (x + y + z)(xy + xz + yz )

-17 = 3xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2

(x + y + z)^3 = 1

x^3 + y^3 + z^3 + 3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = 1

97 + 3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = 1

3(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 6xyz  = -96

(yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2) + 2xyz  = -32

Subtract the following two equations

-17 = 3xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2
-32 = 2xyz + yx^2 + zx^2 + xy^2 + zy^2 + xz^2 + yz^2

15 = xyz

We now have

x + y + z = 1

xy + xz + yz = -17

xyz = 15

So,

(t-x)(t-y)(t-z) = t^3 - t^2 - 17t - 15

The roots of t^3 - t^2 - 17t - 15 are 5 , -1 , -3

So any permutation of 5 , -1 , -3 may be assigned to x , y , z

This gives 6 possible solutions to the given 3 equations

x=5 y=-1 z=-3 ; x=5 y=-3 z=-1 ; x=-1 y=5 z=-3 ; x=-1 y=-3 z=5 ; x=-3 y=-1 z=5 ;x=-3 y=5 z=-1
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks you sir.

Algebra

Volunteer

#### Socrates

##### Expertise

Any questions on High School Algebra, College Algebra, Abstract Algebra. I can help with word problems, solving equations, trigonometry, inequalities, Gaussian Elimination, Linear Algebra, groups, fields, you name it!

##### Experience

Ph.D. in Mathematics, specialist in Algebra. I have taught High School students and college students at three state universities.

Organizations
Mathematical Association of America. American Mathematical Society.

Publications
Regular contributions to the problems section of the American Mathematical Monthly journal.

Education/Credentials
B.S., M.S., Ph.D.