To find \( r \), the variable you want to solve for, depends on the type of equation or formula it is involved in. Here are some common contexts and ways to solve for \( r \):

### 1. **Solving for \( r \) in a Linear Equation**  
   If you have a simple linear equation like \( ar + b = c \), you can solve for \( r \) using the following steps:
   \[
   ar + b = c
   \]
   - Subtract \( b \) from both sides:
   \[
   ar = c - b
   \]
   - Divide both sides by \( a \):
   \[
   r = \frac{c - b}{a}
   \]

### 2. **Solving for \( r \) in a Circle's Equation**  
   If you're given the formula for the circumference or area of a circle, you can solve for the radius \( r \):

   - **Circumference**: \( C = 2\pi r \)
     \[
     r = \frac{C}{2\pi}
     \]

   - **Area**: \( A = \pi r^2 \)
     \[
     r = \sqrt{\frac{A}{\pi}}
     \]

### 3. **Solving for \( r \) in Interest Formulas**  
   In financial mathematics, when using formulas for simple or compound interest, you may need to solve for \( r \):

   - **Simple Interest**: \( I = Prt \) (where \( I \) is interest, \( P \) is principal, \( r \) is the rate, and \( t \) is time)
     \[
     r = \frac{I}{Pt}
     \]

   - **Compound Interest**: \( A = P(1 + r/n)^{nt} \) (where \( A \) is the amount, \( P \) is the principal, \( r \) is the interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is time in years)

### 4. **Solving for \( r \) in the Distance Formula**  
   If you are solving for \( r \) as a distance (e.g., in 3D space):
   - \( d = \sqrt{x^2 + y^2 + z^2} \), where \( d \) is distance
     \[
     r = \sqrt{x^2 + y^2 + z^2}
     \]

### 5. **Solving for \( r \) in the Quadratic Formula**  
   If \( r \) is a root of a quadratic equation \( ax^2 + bx + c = 0 \), use the quadratic formula:
   \[
   r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

If you have a specific formula or context for \( r \), I can help you further with that.

Would you like more details or clarifications? 

Here are 5 related questions for you to explore:
1. How do you solve for \( r \) in the surface area of a sphere formula?
2. What is the role of \( r \) in the Pythagorean theorem?
3. Can you derive the formula for \( r \) in terms of volume for a cylinder?
4. How is the variable \( r \) used in polar coordinates?
5. What are the steps to solve for \( r \) in a logarithmic equation?

**Tip:** Always isolate the variable you're solving for by reversing operations in the equation systematically (addition, subtraction, multiplication, division, powers, etc.).

Discover how to solve for r in various mathematical contexts including linear equations, circles, simple and compound interest formulas, the distance formula, and quadratic equations. Detailed solutions suitable for students in Grades 8-10.

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