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Chapter 5: Problem 13

Solve in complex numbers the following: $$ \left\\{\begin{array}{l} x(x-y)(x-z)=3 \\ y(y-x)(y-z)=3 \\ z(z-x)(z-y)=3 \end{array}\right. $$

Short Answer

Expert verified
Question: Find the solutions for the complex variables x, y, and z in the system of equations: 1. x(x-y)(x-z) - 3 = 0 2. y(y-x)(y-z) - 3 = 0 3. z(z-x)(z-y) - 3 = 0 Answer: The solutions for the complex variables x, y, and z are: x = \frac{3}{(W-y)(W-2y)}, y = \frac{3}{(W-z)(W-x)}, z = \frac{3}{(W-x)(W-y)} where W = \frac{y(y-z)(y-z)}{z(z-y)(z-y)}.

Step by step solution

01

Rewrite the equations to make them easier to manipulate

We can rewrite the given equations as follows: 1. x(x-y)(x-z) - 3 = 0 2. y(y-x)(y-z) - 3 = 0 3. z(z-x)(z-y) - 3 = 0 This makes it easier to manipulate these equations as we try to find the solution for x, y, and z.
02

Sum the first and third equations

Add equation 1 to equation 3 to obtain a new equation: (x(x-y)(x-z) - 3) + (z(z-x)(z-y) - 3) = 0 This equation simplifies to: x(x-y)(x-z) + z(z-x)(z-y) - 6 = 0
03

Sum the first and second equations

Now, add equation 1 to equation 2 to obtain another equation: (x(x-y)(x-z) - 3) + (y(y-x)(y-z) - 3) = 0 This equation simplifies to: x(x-y)(x-z) + y(y-x)(y-z) - 6 = 0
04

Subtract the sum of first and second equations from the sum of first and third equations

Now, subtract the sum of the first and second equations from the sum of the first and third equations to eliminate x from these equations: (x(x-y)(x-z) + z(z-x)(z-y) - 6) - (x(x-y)(x-z) + y(y-x)(y-z) - 6) = 0 The result of this subtraction is: z(z-x)(z-y) - y(y-x)(y-z) = 0
05

Make a substitution to further simplify the equation

Introduce a new variable, W: W = x - y We can then rewrite the equation we got in step 4 as: z(z-W)(z-y) - y(y-W)(y-z) = 0
06

Solve for W

Now we can solve for W: z(z-W)(z-y) = y(y-W)(y-z) Divide both sides by -(z-y): W = \frac{y(y-z)(y-z)}{z(z-y)(z-y)} Now we have an expression for W = x - y
07

Substitute the expression for W back into the first equation and solve for x

Substitute the expression for W back into equation 1: x((x-y) - y) (x - (y + W)) - 3 = 0 Now we can solve for x: x = \frac{3}{(W-y)(W-2y)}
08

Substitute the expressions for x and W back into the second equation and solve for y

Substitute the expressions for x and W back into equation 2: y\left(y-\frac{3}{(W-y)(W-2y)}\right)(y - (z + x)) - 3 = 0 Now we can solve for y: y = \frac{3}{(W-z)(W-x)}
09

Substitute the expressions for x, y and W back into the third equation and solve for z

Substitute the expressions for x, y, and W back into equation 3: z\left(z - \frac{3}{(W-y)(W-2y)}\right)(z - (3y - W)) - 3 = 0 Now we can solve for z: z = \frac{3}{(W-x)(W-y)} Finally, with the expressions for x, y, and z found, the solutions for the complex variables x, y, and z are: x = \frac{3}{(W-y)(W-2y)}, y = \frac{3}{(W-z)(W-x)}, z = \frac{3}{(W-x)(W-y)}

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