Let's solve the equation step by step.

The given equation is:

\[
\frac{1}{z} - \frac{1}{2z} - \frac{1}{5z} = \frac{10}{z} + 1
\]

### Step 1: Combine the terms on the left-hand side
All the terms on the left-hand side have a common denominator of \( z \). Let's rewrite the equation:

\[
\frac{1}{z} - \frac{1}{2z} - \frac{1}{5z} = \frac{10}{z} + 1
\]

To combine the fractions on the left-hand side, factor out \( \frac{1}{z} \):

\[
\frac{1}{z} \left( 1 - \frac{1}{2} - \frac{1}{5} \right) = \frac{10}{z} + 1
\]

### Step 2: Simplify the terms inside the parentheses
Now, let's find a common denominator for the fractions inside the parentheses:

\[
1 - \frac{1}{2} - \frac{1}{5} = \frac{10}{10} - \frac{5}{10} - \frac{2}{10} = \frac{3}{10}
\]

So, the equation becomes:

\[
\frac{1}{z} \cdot \frac{3}{10} = \frac{10}{z} + 1
\]

### Step 3: Simplify the equation
Now the equation is:

\[
\frac{3}{10z} = \frac{10}{z} + 1
\]

Multiply both sides of the equation by \( 10z \) to eliminate the denominators:

\[
3 = 100 + 10z
\]

### Step 4: Solve for \( z \)
Now, subtract 100 from both sides:

\[
3 - 100 = 10z
\]

\[
-97 = 10z
\]

Now, divide by 10:

\[
z = \frac{-97}{10}
\]

Thus,

Explore how to solve the fractional linear equation 1/z - 1/2z - 1/5z = 10/z + 1 with a comprehensive step-by-step guide. This problem involves combining fractions, simplifying expressions, and solving for a variable, making it ideal for students in Grades 8-10.

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