To solve the given problem, we need to determine the value of $m$ for which the solutions of the two equations are reciprocals of each other, and then find the value of $\left( m + \frac{1}{2} \right)^3$.

Let's break down the problem step-by-step:

Step 1: Understand the Equations

We have two equations:

1. $3(x - 1) = 3m - 6$
2. $2x - 5 = -1$

Step 2: Simplify the Equations

Let's simplify each equation to the standard form $x =$.

For the first equation: $3(x - 1) = 3m - 6$ $3x - 3 = 3m - 6$ $3x = 3m - 3$ $x = m - 1$

For the second equation: $2x - 5 = -1$ $2x = 4$ $x = 2$

Step 3: Determine the Relationship Between the Solutions

We need the solutions $x = m - 1$ and $x = 2$ to be reciprocals of each other. This means: $(m - 1) \cdot 2 = 1$

Step 4: Solve for $m$

$2(m - 1) = 1$ $2m - 2 = 1$ $2m = 3$ $m = \frac{3}{2}$

Step 5: Calculate the Required Expression

Now, we need to find the value of $\left( m + \frac{1}{2} \right)^3$: $m = \frac{3}{2}$ $m + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = 2$ $\left( 2 \right)^3 = 8$

So, the value of $\left( m + \frac{1}{2} \right)^3$ is 8.

Final Answer

$\left( m + \frac{1}{2} \right)^3 = 8$

Would you like any more details or have any questions about this solution?

Here are 5 numbered questions you might want to ask:

1. How do you solve a system of linear equations?
2. What does it mean for two numbers to be reciprocals?
3. How do you simplify algebraic expressions?
4. Can you explain the properties of exponents?
5. What are the steps to solving a quadratic equation?

Tip: When solving equations, always simplify each equation as much as possible to isolate the variable. This makes it easier to identify relationships and solve for the unknowns.