What is the justification for step 2 in the solution process?\begin{array} {rcrclC40C40C40C40C40}&&\frac{1}{2}r\:+\:\frac{1}{2}&=&\textrm{-}\frac{2}{7}r\:+\:\frac{6}{7}\:-\:5\\ \textrm{Step 1}&:&\frac{1}{2}r\:+\:\frac{1}{2}&=&\textrm{-}\frac{2}{7}r\:-\:\frac{29}{7}\\ \textrm{Step 2}&:&\frac{1}{2}r&=&\textrm{-}\frac{2}{7}r\:-\:\frac{65}{14} \end{array}

The justification for Step 2 in the solution process involves isolating the term with the variable \( r \) on the left-hand side of the equation by eliminating the constant term on the left-hand side.

### Here's how it works:

1. **Starting from Step 1:**
   \[
   \frac{1}{2}r + \frac{1}{2} = -\frac{2}{7}r - \frac{29}{7}
   \]

2. **Subtract \(\frac{1}{2}\) from both sides:**
   To eliminate the constant term \(\frac{1}{2}\) on the left-hand side, subtract \(\frac{1}{2}\) from both sides of the equation:
   \[
   \frac{1}{2}r + \frac{1}{2} - \frac{1}{2} = -\frac{2}{7}r - \frac{29}{7} - \frac{1}{2}
   \]
   Simplifying this gives:
   \[
   \frac{1}{2}r = -\frac{2}{7}r - \left(\frac{29}{7} + \frac{1}{2}\right)
   \]

3. **Combine the constants on the right-hand side:**
   The right-hand side now involves adding \(\frac{29}{7}\) and \(\frac{1}{2}\) as follows:
   \[
   -\frac{29}{7} - \frac{1}{2} = -\frac{58}{14} - \frac{7}{14} = -\frac{65}{14}
   \]
   So the equation becomes:
   \[
   \frac{1}{2}r = -\frac{2}{7}r - \frac{65}{14}
   \]

This is Step 2 in the solution process. The key justification for Step 2 is subtracting \(\frac{1}{2}\) from both sides to isolate the term containing \( r \) on the left side.

Would you like more details or have any further questions?

### Related Questions:
1. How do you combine fractions with different denominators?
2. What is the next step after isolating the variable terms on each side?
3. How do you solve equations with variables on both sides?
4. What does it mean to isolate a variable in an equation?
5. How do you simplify complex fractions?

**Tip:** Always check that the fractions are simplified before combining them.

Understand how to solve equations involving fractions step-by-step. This example covers the process of isolating variables and combining fractions in equations. Suitable for high school students.

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