Given \( C = 31.4 \), solve for \( r \): \( 2 \times 3.14 \times r = 31.4 \).
["Solving for ( r ) in the Equation ( 2 \ imes 3.14 \ imes r = 31.4 ): A Clear Step-by-Step Guide", "Understanding how to solve for a variable in a simple linear equation is essential in math, science, and real-world applications. In this article, we’ll solve the equation ( 2 \ imes 3.14 \ imes r = 31.4 ) for ( r ), using the known value ( C = 31.4 ) and confirmed constant ( 2 \ imes 3.14 ). This example illustrates core algebra principles that apply across many problem types.", "---", "### The Equation: Why It Matters", "The equation ( 2 \ imes 3.14 \ imes r = 31.4 ) represents a proportional relationship. Here:", "- ( 2 \ imes 3.14 ) is a constant multiplier (approximately ( C )),\n- ( r ) is the unknown variable we want to isolate,\n- ( 31.4 ) is the known total value.", "By solving for ( r ), we uncover how many times this constant multiplier fits into the total—key in physics, engineering, statistics, and economics.", "---", "### Step 1: Simplify the left-hand side", "First, compute the constant factor:", "[\n2 \ imes 3.14 = 6.28\n]", "So the equation becomes:", "[\n6.28 \ imes r = 31.4\n]", "---", "### Step 2: Isolate ( r ) by dividing both sides by 6.28", "To solve for ( r ), divide both sides of the equation by ( 6.28 ):", "[\nr = \frac{31.4}{6.28}\n]", "---", "### Step 3: Perform the division", "Carry out the calculation:", "[\nr = \frac{31.4}{6.28} = 5\n]", "Alternatively, recognizing ( 6.28 = 2 \ imes 3.14 = C ), the equation simplifies neatly:", "[\nr = \frac{C}{2 \ imes 3.14} = \frac{31.4}{6.28} = 5\n]", "---", "### Final Result", "[\n\boxed{r = 5}\n]", "---", "### Why This Formula Works and How to Use It", "This method reflects the rearrangement property of equality in algebra:", "[\na \ imes r = b \quad \Rightarrow \quad r = \frac{b}{a}\n]", "Here, ( a = 6.28 ) and ( b = 31.4 ), making the solution direct and reliable. Whether you're calculating rates, scaling formulas, or analyzing proportional systems, this algebraic approach delivers clarity and precision.", "---", "### Summary", "- Always simplify constants first.\n- Use division to isolate the unknown variable.\n- Recognizing known values (like ( \pi \approx 3.14 )) helps streamline equations.\n- Verifying with substitution confirms correctness: plug ( r = 5 ) back into ( 6.28 \ imes 5 = 31.4 ), which checks true.", "By mastering such equations, you build a strong foundation in quantitative reasoning—essential for academic success and real-world problem solving.", "---", "Keywords for SEO: solve for r, equation solving, linear algebra, algebra practice, solve 6.28 × r = 31.4, step-by-step math guide, real-world applications, r value calculation, simplify variables, math equations explained."]
