# Prove Algebra Challenge: $(x,y,z,a,b,c)$ Equation

• MHB
• anemone
In summary, for reals $x,\,y,\,z$ and $a,\,b$ and $c$ that satisfy $a + b + c = ax + by + cz = x^2a + y^2b + z^2c = 1$, it is proven that $x^3a + y^3b + z^3c = 1 − (1 − x)(1 − y)(1 − z)$.
anemone
Gold Member
MHB
POTW Director
For reals $x,\,y,\,z$ and $a,\,b$ and $c$ that satisfy $a + b + c = ax + by + cz = x^2a + y^2b + z^2c = 1$,

prove that $x^3a + y^3b + cz^3c = 1 − (1 − x)(1 − y)(1 − z)$

anemone said:
For reals $x,\,y,\,z$ and $a,\,b$ and $c$ that satisfy $a + b + c = ax + by + cz = x^2a + y^2b + z^2c = 1$,

prove that $x^3a + y^3b + cz^3c = 1 − (1 − x)(1 − y)(1 − z)$

I think you mean
prove that $x^3a + y^3b + z^3c = 1 − (1 − x)(1 − y)(1 − z)$

we are given
$a+b+c=\cdots(1)$
$ax+by+cz=1\cdots(2)$
$ax^2+by^2+cz^2 = 1\cdots(3)$
so we have
$x^3a+y^3b+z^3c$
= $x( 1- y^2b-z^2c) + y (1-x^2a-z^2c) + z(1-x^2a-y^2b)$ using (3)
= $x+y+z- xy(by+ax) - zx(cz+ax) - yz(by+cz)$
= $x+y+z-xy(1-cz) - zx(1-by) - yz(1-ax)$ using (2) in each of 3 expressions
= $x+y+z - xy - zx - yz + xyz(c+b+a)$
= $x+y+z - xy - zx - yz + xyz$ using (1)
= $x-xy-xz +xyz + y + z - yz$
= $x(1-y-z+yz) + (y+z-yz)$
=$x(1-y)(1-z) - (1-y)(1-z)+1$
= $1 + (x-1)(1-y)(1-z)$
= $1- (1-x)(1-y)(1-z)$

Last edited:
I think you mean
prove that $x^3a + y^3b + z^3c = 1 − (1 − x)(1 − y)(1 − z)$

we are given
$a+b+c=\cdots(1)$
$ax+by+cz=1\cdots(2)$
$ax^2+by^2+cz^2 = 1\cdots(3)$
so we have
$x^3a+y^3b+z^3c$
= $x( 1- y^2b-z^2c) + y (1-x^2a-z^2c) + z(1-x^2a-y^2b)$ using (3)
= $x+y+z- xy(by+ax) - zx(cz+ax) - yz(by+cz)$
= $x+y+z-xy(1-cz) - zx(1-by) - yz(1-ax)$ using (2) in each of 3 expressions
= $x+y+z - xy - zx - yz + xyz(c+b+a)$
= $x+y+z - xy - zx - yz + xyz$ using (1)
= $x-xy-xz +xyz + y + z - yz$
= $x(1-y-z+yz) + (y+z-yz)$
=$x(1-y)(1-z) - (1-y)(1-z)+1$
= $1 + (x-1)(1-y)(1-z)$
= $1- (1-x)(1-y)(1-z)$

## 1) What is the purpose of the "Prove Algebra Challenge" equation?

The purpose of the equation is to test one's understanding and application of algebraic principles and concepts. It challenges individuals to prove the given equation using various mathematical techniques and strategies.

## 2) Can you provide an example of the "Prove Algebra Challenge" equation?

An example of the equation could be something like $(x+y)^2=x^2+y^2+2xy$. This equation requires individuals to prove the given statement using algebraic manipulation and substitution.

## 3) What level of algebra is required to solve the "Prove Algebra Challenge" equation?

The level of algebra required can vary depending on the specific equation given. Generally, it may require knowledge of basic algebraic operations, such as simplifying expressions, solving equations, and using properties of exponents and polynomials.

## 4) How long do individuals typically take to solve the "Prove Algebra Challenge" equation?

The time it takes to solve the equation can vary greatly depending on an individual's level of algebraic understanding and problem-solving skills. It could take anywhere from a few minutes to several hours, or even longer for more complex equations.

## 5) Is there a specific method or approach to solving the "Prove Algebra Challenge" equation?

There is no specific method or approach that must be used to solve the equation. Individuals may use various techniques, such as algebraic manipulation, substitution, factoring, or properties of exponents, to prove the given equation. The key is to understand the principles involved and apply them correctly to reach a logical solution.

• General Math
Replies
1
Views
909
• General Math
Replies
1
Views
1K
• General Math
Replies
4
Views
1K
• Precalculus Mathematics Homework Help
Replies
3
Views
1K
• General Math
Replies
19
Views
2K
• General Math
Replies
2
Views
1K
• General Math
Replies
1
Views
955
• Calculus and Beyond Homework Help
Replies
2
Views
693
• General Math
Replies
10
Views
2K
• General Math
Replies
1
Views
765