Cross-multiply to solve for \( y \):

["# Solve for ( y ) Using Cross-Multiplication: A Simple & Efficient Method", "When solving equations involving fractions, one of the most effective techniques for isolating the variable is cross-multiplication. This method streamlines the process of solving linear equations and is especially useful in algebra 1, precalc, and high school math. In this article, we’ll explain how cross-multiplication works, why it’s powerful, and walk you through solving for ( y ) step by step.", "---", "## What Is Cross-Multiplication?", "Cross-multiplication is a simple algebraic method used to eliminate fractions by multiplying both the numerator of one fraction by the denominator of the other, and vice versa. It’s based on the principle that if two ratios are equal, their cross products are equal.", "Mathematically, if\n[\n\frac{a}{b} = \frac{c}{d}\n]\nthen cross-multiplying gives:\n[\na \cdot d = b \cdot c\n]", "This transforms a proportional equation into a linear equation—perfect for solving for unknown variables like ( y ).", "---", "## Why Use Cross-Multiplication?", "- Eliminates fractions quickly without needing to find common denominators.\n- Simplifies complex expressions into straightforward multiplication.\n- Minimizes errors common with traditional fraction solving.\n- Works for any ratio equality, not just simple numbers—great for variables too.", "---", "## How to Solve for ( y ) Using Cross-Multiplication", "Let’s solve a typical equation step by step using cross-multiplication.", "### Step 1: Identify the structure", "Suppose we have an equation like:\n[\n\frac{3y + 2}{5} = \frac{y - 4}{2}\n]", "Here, the variable ( y ) appears on both sides, and both sides are fractions involving ( y ).", "### Step 2: Cross-multiply", "Multiply the numerator of the left by the denominator of the right, and vice versa:\n[\n(3y + 2) \cdot 2 = 5 \cdot (y - 4)\n]", "### Step 3: Expand both sides\nLeft side:\n[\n2(3y + 2) = 6y + 4\n]\nRight side:\n[\n5(y - 4) = 5y - 20\n]", "So now the equation is:\n[\n6y + 4 = 5y - 20\n]", "### Step 4: Isolate ( y )", "Subtract ( 5y ) from both sides:\n[\n6y - 5y + 4 = -20\n]\n[\ny + 4 = -20\n]", "Subtract 4 from both sides:\n[\ny = -20 - 4\n]\n[\ny = -24\n]", "### Step 5: Check the solution", "Plug ( y = -24 ) back into the original equation to confirm:", "Left:\n[\n\frac{3(-24) + 2}{5} = \frac{-72 + 2}{5} = \frac{-70}{5} = -14\n]", "Right:\n[\n\frac{-24 - 4}{2} = \frac{-28}{2} = -14\n]", "Both sides equal ( -14 ), so the solution is valid.", "---", "## Real-World Applications", "Cross-multiplying to solve for ( y ) isn’t just academic—it’s practical. Engineers, scientists, and computer programmers regularly use this method to solve proportional relationships, optimize systems, and model real-world phenomena.", "---", "## Tips for Mastering Cross-Multiplication", "- Always double-check signs when expanding both sides.\n- Watch out for parentheses—apply the distributive property correctly.\n- After isolating ( y ), verify solutions by substitution.\n- Practice with variables on both sides to build confidence.", "---", "## Conclusion", "Cross-multiplication is a powerful, elegant tool for solving equations with fractions and variables. By converting ratios into multiplicative equality, it accelerates the solving process and reduces complexity. Whether you're solving equations in class, on tests, or in professional settings, mastering this technique will strengthen your algebraic foundation and boost confidence in math.", "Start practicing with simple equations, and soon you’ll solve for ( y )—and more—with speed and precision.", "---", "Keywords: cross-multiply, solve for y, algebra, solving linear equations, fractions algebra, step-by-step solution, equation manipulation, math tips, solving proportions", "Meta Description: Learn how to solve for ( y ) using cross-multiplication with step-by-step examples, tips, and real-world applications. Master algebra quickly and confidently."]