How to Solve a Transformation Problem for Variables a and b?

[LakeMountD]

Find a & b for the following transformation

1/(z+a) = (a/z) + [b/(1-z)]

I am kind of lost as to where to start on this and google didn't help much typing in math transformations. I know how to do laplace transforms and such but not really sure what they are asking here and we aren't to Laplace yet.

 

[Tide]

Why don't you just set z to two different values then solve the resulting pair of equations for a and b?

 

[tacman]

The transformation problem in this context refers to finding values for the variables a and b that will satisfy the given transformation equation. In this case, the equation is 1/(z+a) = (a/z) + [b/(1-z)]. To solve for a and b, we can use algebraic manipulation and properties of fractions.

First, let's combine the fractions on the right side of the equation by finding a common denominator. This will give us:

1/(z+a) = (a(1-z) + bz)/(z(1-z))

Next, we can expand the numerator on the right side:

1/(z+a) = (a-a*z + bz)/(z-z^2)

We can then simplify the numerator by factoring out a common factor of a, giving us:

1/(z+a) = a(1-z+b)/(-z^2)

Now, we can equate the numerators on both sides of the equation, giving us:

1 = a(1-z+b)

We can then expand the right side and rearrange the terms to get:

1 = (a-a*z+ab)

Next, we can equate the coefficients of z on both sides of the equation, giving us:

0 = -a + b

Finally, we can substitute this value for b into our original equation and solve for a:

1 = (a-a*z+a-a)

1 = (2a-a*z)

1 = a(2-z)

a = 1/(2-z)

So, we have found a value for a in terms of z. To find a value for b, we can substitute this value of a into our equation 0 = -a + b and solve for b:

0 = -1/(2-z) + b

b = 1/(2-z)

Therefore, the values of a and b that satisfy the given transformation equation are:

a = 1/(2-z)

b = 1/(2-z)

In summary, the transformation problem involves finding values for the variables a and b that will satisfy the given transformation equation. In this case, we used algebraic manipulation and properties of fractions to solve for a and b in terms of z.