Solve: (3z/4 + 1/4m + 12/13z⋅4/5m + 1/7z) Combined Expression

To solve this problem, we'll follow these steps:

• Identify and combine like terms involving $z$.

• Simplify the term involving both $z$ and $m$.

• Combine and simplify fractions.

Now, let us solve the expression step-by-step:

Given expression: $\frac{3z}{4} + \frac{1}{4}m + \frac{12}{13}z \cdot \frac{4}{5}m + \frac{1}{7}z$

Step 1: Combine like terms for $z$.

Terms involving $z$ are: $\frac{3z}{4}$, $\frac{1}{7}z$.

To combine these, we need a common denominator. The least common multiple of 4 and 7 is 28:

$\frac{3z}{4} = \frac{21z}{28}$ and $\frac{1z}{7} = \frac{4z}{28}$.

Add these: $\frac{21z}{28} + \frac{4z}{28} = \frac{25z}{28}$.

Step 2: Simplify the term involving both $z$ and $m$.

$\frac{12}{13}z \cdot \frac{4}{5}m = \frac{48}{65}zm$.

This expression is already in its simplest form.

Step 3: Write the whole expression in simplified form:

$\frac{25z}{28} + \frac{1}{4}m + \frac{48}{65}zm$.

Therefore, the simplified expression is: $\frac{25}{28}z + \frac{1}{4}m + \frac{48}{65}zm$.