Let's solve the given equation step-by-step:
Given:
[tex]\[
\frac{3z - m}{5z - n} = \frac{3z + n}{5z - m}
\][/tex]
### Step 1: Cross-multiply
Cross-multiplication helps to eliminate the fractions:
[tex]\[
(3z - m)(5z - m) = (3z + n)(5z - n)
\][/tex]
### Step 2: Expand both sides
Expand both sides of the equation by distributing the terms:
[tex]\[
(3z - m)(5z - m) = 3z \cdot 5z - 3z \cdot m - m \cdot 5z + m \cdot m = 15z^2 - 3zm - 5zm + m^2 = 15z^2 - 8zm + m^2
\][/tex]
[tex]\[
(3z + n)(5z - n) = 3z \cdot 5z - 3z \cdot n + n \cdot 5z - n \cdot n = 15z^2 - 3zn + 5zn - n^2 = 15z^2 + 2zn - n^2
\][/tex]
So, we have:
[tex]\[
15z^2 - 8zm + m^2 = 15z^2 + 2zn - n^2
\][/tex]
### Step 3: Simplify the equation
Subtract [tex]\(15z^2\)[/tex] from both sides to simplify:
[tex]\[
-8zm + m^2 = 2zn - n^2
\][/tex]
### Step 4: Rearrange the terms to solve for [tex]\(z\)[/tex]
Bring all the terms involving [tex]\(z\)[/tex] on one side and the constants on the other side:
[tex]\[
-8zm - 2zn = -n^2 - m^2
\][/tex]
Factor out the common term [tex]\(z\)[/tex] on the left side:
[tex]\[
z(-8m - 2n) = -n^2 - m^2
\][/tex]
### Step 5: Solve for [tex]\(z\)[/tex]
Divide both sides by [tex]\(-8m - 2n\)[/tex]:
[tex]\[
z = \frac{n^2 + m^2}{8m + 2n}
\][/tex]
To simplify further, factor out the common factor in the denominator:
[tex]\[
z = \frac{n^2 + m^2}{2(4m + n)}
\][/tex]
Thus, the solution for [tex]\(z\)[/tex] is:
[tex]\[
z = \frac{m^2 + n^2}{2(4m + n)}
\][/tex]
So, the answer is:
[tex]\[
z = \frac{m^2 + n^2}{2(4m + n)}
\][/tex]
