Add the equations: \( 2x = 13 + \frac{45}{13} = \frac{169 + 45}{13} = \frac{214}{13} \).

["Understanding the Equation: Step-by-Step Breakdown of ( 2x = 13 + \frac{45}{13} = \frac{169 + 45}{13} = \frac{214}{13} )", "Solving algebraic equations is a fundamental skill in mathematics, and breaking equations down step by step can help enhance comprehension and accuracy. One such expression—( 2x = 13 + \frac{45}{13} )—can be simplified and rewritten using clear mathematical steps, ultimately arriving at the elegant form ( \frac{214}{13} ). In this article, we’ll explore how to derive this result with detailed explanations and practical applications of algebraic manipulation.", "---", "### Step 1: Start with the Original Equation", "We begin with:\n[ 2x = 13 + \frac{45}{13} ]", "This equation expresses ( 2x ) as the sum of two terms: a whole number ( 13 ) and a fraction ( \frac{45}{13} ).", "---", "### Step 2: Combine the Terms on the Right Side", "To combine ( 13 + \frac{45}{13} ), we express ( 13 ) as a fraction with denominator 13:", "[\n13 = \frac{13 \ imes 13}{13} = \frac{169}{13}\n]", "So now the right-hand side becomes:\n[\n2x = \frac{169}{13} + \frac{45}{13}\n]", "---", "### Step 3: Add the Fractions", "Since the denominators are identical, we can directly add the numerators:", "[\n2x = \frac{169 + 45}{13} = \frac{214}{13}\n]", "---", "### Step 4: Solve for ( x )", "To isolate ( x ), divide both sides by 2:", "[\nx = \frac{214}{13} \div 2 = \frac{214}{13} \ imes \frac{1}{2} = \frac{214}{26} = \frac{107}{13}\n]", "Thus, ( x = \frac{107}{13} ), but the key step to recognize is how the fraction simplified into a clean form.", "---", "### Why This Simplification Matters", "Reporing an expression as ( 2x = \frac{214}{13} ) reveals it as a linear equation with a clear, rational right-hand side—useful in solving real-world problems, graphing lines, or simplifying further expressions. Algebraic identities like this help avoid errors and clarify which operations are involved.", "---", "### Visual Representation: A Numerical Check", "Let’s verify:\n[\n2x = \frac{214}{13} \Rightarrow x = \frac{107}{13} \approx 8.23\n]\nNow compute original RHS numerically:\n[\n13 + \frac{45}{13} = 13 + 3.4615 \approx 16.4615 = \frac{169 + 45}{13} = \frac{214}{13} \approx 16.4615\n]\nBoth sides match! This confirms the validity of the step-by-step transformation.", "---", "### Applications and Extensions", "Understanding how to combine fractions and isolate variables opens doors to solving complex equations, working with rational numbers, and modeling situations requiring proportional reasoning—such as scaling recipes, calculating rates, or analyzing financial ratios.", "---", "### Conclusion", "Simplifying ( 2x = 13 + \frac{45}{13} ) step by step—first by rewriting integers as fractions, then combining them, and finally isolating ( x )—illustrates the power of foundational algebra. The result ( \frac{214}{13} ) not only answers the equation but also serves as a crucial building block in higher mathematics. Whether studying algebra, preparing for exams, or applying math in practical fields, mastering such transformations is essential.", "For further practice, try combining fractions with different denominators or solving similar equations—each step strengthens your algebraic intuition and problem-solving confidence.", "---", "Keywords:\nalgebra, solving equations, fraction arithmetic, algebraic simplification, solving linear equations, ( 2x = \frac{214}{13} ), fraction addition, rational expressions, mathematical steps, algebra tutorial", "Meta Description:\nLearn how to simplify ( 2x = 13 + \frac{45}{13} ) into ( \frac{214}{13} ) step-by-step. Understand fraction addition, combine like terms, and solve for ( x )—essential skills in algebra and beyond."]