Substitute back to find \( y \): \( y = 13 - \frac{107}{13} = \frac{169 - 107}{13} = \frac{62}{13} \).
["# Solving for ( y ) in the Equation: ( y = 13 - \frac{107}{13} )", "Understanding how to isolate and solve for a variable is a fundamental skill in algebra. One common task is rewriting an expression like ( y = 13 - \frac{107}{13} ) into an equivalent fraction form—especially useful when simplifying or comparing values. This article shows step-by-step how to find ( y ) by evaluating and simplifying the expression ( y = 13 - \frac{107}{13} ), arriving at ( y = \frac{62}{13} ).", "---", "## The Original Expression", "We start with the expression:", "[\ny = 13 - \frac{107}{13}\n]", "This equation tells us ( y ) is composed of two terms: a whole number (13) and a fraction (-\frac{107}{13}). To simplify, we convert 13 into a fraction with denominator 13 so terms can be combined.", "---", "## Step 1: Convert 13 to a Fraction with Denominator 13", "Since ( 13 = \frac{13}{1} ), we rewrite it with denominator 13:", "[\n13 = \frac{13 \ imes 13}{1 \ imes 13} = \frac{169}{13}\n]", "Now the equation becomes:", "[\ny = \frac{169}{13} - \frac{107}{13}\n]", "---", "## Step 2: Subtract the Fractions", "Since the denominators are the same, subtract the numerators directly:", "[\ny = \frac{169 - 107}{13}\n]", "Calculate the numerator:", "[\n169 - 107 = 62\n]", "So:", "[\ny = \frac{62}{13}\n]", "---", "## Final Simplified Value", "Thus, solving for ( y ):", "[\ny = 13 - \frac{107}{13} = \frac{62}{13}\n]", "This fraction is already in simplest form because 62 and 13 share no common factors other than 1.", "---", "## Why This Simplification Matters", "Expressing ( y ) as ( \frac{62}{13} ) provides a concise and exact representation of the value. It’s easier to work with in further calculations, graphing, or comparing against other numerical solutions. Converting mixed numbers back to improper fractions ensures clarity and precision.", "---", "## Practice: Replacing ( y ) in Equations", "To reinforce your understanding, try solving similar expressions. For example:", "- Express ( 25 - \frac{71}{25} ) as a single fraction.\n- Simplify ( 8 - \frac{47}{8} ) and convert to an improper fraction.", "%% Additional formula reminder:\n[\na - \frac{b}{a} = \frac{a^2 - b}{a}\n]", "Use this pattern anytime you subtract a fraction from a whole number expressed as ( \frac{a}{1} ).", "---", "## Conclusion", "Solving ( y = 13 - \frac{107}{13} ) simplifies neatly to ( y = \frac{62}{13} ), highlighting the power of common denominators in algebra. Mastering such steps ensures smooth progress in solving equations and mastering rational expressions. Whether for homework, tests, or real-world applications, efficient fractional simplification is essential.", "---", "### Key Takeaways:\n- Always convert whole numbers to fractions to combine terms.\n- Subtract fractions by keeping the same denominator and combining numerators.\n- Simplify the result if possible.\n- Using the formula ( a - \frac{b}{a} = \frac{a^2 - b}{a} ) streamlines work.", "Optimize your algebra practice by applying these techniques consistently!"]
