https://file.aiquickdraw.com/mathbot/file/17264481809814o9g9m2w.jpg

The equation you provided is:

\[
-\frac{22}{5}z = -\frac{23}{10}(2z + 1)
\]

Let's solve for \( z \) step by step.

### Step 1: Expand the right-hand side
Distribute \(-\frac{23}{10}\) over \(2z + 1\):

\[
-\frac{23}{10}(2z + 1) = -\frac{23}{10} \cdot 2z - \frac{23}{10} \cdot 1
\]
\[
= -\frac{46}{10}z - \frac{23}{10}
\]
\[
= -\frac{23}{5}z - \frac{23}{10}
\]

Now the equation becomes:
\[
-\frac{22}{5}z = -\frac{23}{5}z - \frac{23}{10}
\]

### Step 2: Eliminate the fractions
Multiply the entire equation by 10 to get rid of the denominators:

\[
10 \times \left(-\frac{22}{5}z\right) = 10 \times \left(-\frac{23}{5}z - \frac{23}{10}\right)
\]

\[
-44z = -46z - 23
\]

### Step 3: Solve for \( z \)
Add \( 46z \) to both sides:

\[
-44z + 46z = -23
\]

\[
2z = -23
\]

Now divide both sides by 2:

\[
z = -\frac{23}{2}
\]

So, the solution is:

\[
z = -\frac{23}{2}
\]

Would you like a detailed explanation of any step? Here are 5 related questions to deepen your understanding:

1. How do you handle equations with fractions?
2. Why do we multiply both sides by 10 in Step 2?
3. How can you check the solution for \( z \) in the original equation?
4. What happens if you have variables on both sides of the equation?
5. Can this type of equation have more than one solution?

**Tip:** Always remember to simplify your equation as much as possible before solving for the variable!

Learn how to solve the linear equation -22/5z = -23/10(2z + 1) with fractions and variables on both sides. This problem covers key algebraic concepts like the distributive property and solving equations with fractions, suitable for students in grades 9-10.

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