Let's solve the given expression step by step.

The expression is:

\[
r = \frac{1}{4}(3r - 2) - 7 + \frac{3}{4}r
\]

### Step 1: Distribute the \(\frac{1}{4}\) across \((3r - 2)\)

\[
r = \frac{1}{4} \cdot 3r - \frac{1}{4} \cdot 2 - 7 + \frac{3}{4}r
\]
\[
r = \frac{3}{4}r - \frac{2}{4} - 7 + \frac{3}{4}r
\]
\[
r = \frac{3}{4}r - \frac{1}{2} - 7 + \frac{3}{4}r
\]

### Step 2: Combine like terms involving \(r\)

\[
r = \left(\frac{3}{4}r + \frac{3}{4}r\right) - \frac{1}{2} - 7
\]
\[
r = \frac{6}{4}r - \frac{1}{2} - 7
\]
\[
r = \frac{3}{2}r - \frac{1}{2} - 7
\]

### Step 3: Combine constants

\[
r = \frac{3}{2}r - \left( \frac{1}{2} + 7 \right)
\]
\[
r = \frac{3}{2}r - \frac{15}{2}
\]

### Step 4: Eliminate the fraction by multiplying the entire equation by 2

\[
2r = 3r - 15
\]

### Step 5: Solve for \(r\)

Move all terms involving \(r\) to one side:

\[
2r - 3r = -15
\]
\[
-r = -15
\]

Divide by -1:

\[
r = 15
\]

### Final Answer: \(r = 15\)

Would you like more details, or do you have any questions?

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Here are 5 related questions to further explore this topic:

1. How do you solve an equation with fractions on both sides?
2. What happens if you multiply both sides of an equation by a negative number?
3. How do you check if your solution to an equation is correct?
4. What are the different methods for solving linear equations?
5. How do you solve an equation where the variable appears in multiple terms?

**Tip**: Always simplify equations by combining like terms and removing fractions early in the process.

This page provides a step-by-step solution to the linear equation r = 1/4(3r - 2) - 7 + 3/4r. Key algebraic concepts such as the distributive property, combining like terms, and working with fractions are explained. Suitable for students in Grades 8-10.

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