Solving for x - z Given xyz a and x2y3z b

In mathematical tasks, solving for variables given specific equations can be quite challenging. Let's explore how to find the value of x - z when we have the following two equations:

Equations and Initial Setup

#8711; #8711; #8711; a
x #8711; 2y #8711; 3z b

Let's denote these equations as:

x y z a (Equation 1) x 2y 3z b (Equation 2)

Our goal is to find the value of x - z. This involves a series of equations to isolate and substitute variables.

Manipulating Equations to Isolate Variables

To solve for x, y, and z, we need to manipulate the equations step by step. Starting with Equation 1 and Equation 2:

Subtracting Equation 1 from Equation 2

Equation 1: x y z a

Equation 2: x 2y 3z b

Subtracting Equation 1 from Equation 2:

x 2y 3z - x y z b - a

This simplifies to:

y 2z b - a

Expressing y from Equation 1

From Equation 1: y a - x - z

Substituting y into the Simplified Equation 2

Substituting the expression for y into the simplified Equation 2:

a - x - z 2z b - a

Simplifying:

a - x z b - a

Rearranging:

- x z b - a - a

- x z b - 2a

Expressing z in Terms of x

From the above simplification, we can express z in terms of x:

z x (b - 2a)

Substituting z Back to Find x - z

Substituting the expression for z back into x - z:

x - z x - x (b - 2a)

This simplifies to:

x - z - (b - 2a)

x - z 2a - b

Thus, the final result is:

boxed{2a - b}

Verification and Alternative Methods

Let's verify the solution using alternative methods:

Verification 1

Given:

xyz a x2y3z b

We can express y 2z b - a as:

y (b - a) / 2z

Substituting z a - x - y into the equation:

x - (a - x - y) 2a - b

Verification 2

Given:

x y z a x 2y 3z b

Subtracting the two equations:

x 2y 3z - x y z b - a

x - z b - 2a

Thus, x - z 2a - b.

Conclusion

We have demonstrated step-by-step solutions to find the value of x - z given the equations x y z a and x 2y 3z b. The final answer is: