Cross-multiplying: 2(5x - 4) = 7x + 2 → 10x - 8 = 7x + 2 → 3x = 10 → x = 10/3.

["Cross-Multiplying Methods Simplified: How to Solve Linear Equations Step-by-Step", "Solving linear equations is a fundamental skill in algebra that opens the door to advanced math concepts. One common technique involved in many equations is cross-multiplication, especially when dealing with proportions or expressions involving variables. In this article, we’ll explore how cross-multiplying applies to solving equations like 2(5x – 4) = 7x + 2, guiding high school students and self-learners through the process clearly and effectively.", "---", "### Understanding Cross-Multiplying: More Than Just Fractions", "Though cross-multiplying is often introduced in the context of fractions (like solving (\frac{a}{b} = \frac{c}{d}) by multiplying both sides by (b \cdot d)), this method also plays a key role in solving equations where variables appear on both sides. In essence, cross-multiplying helps eliminate denominators or simplify complex expressions to isolate variables cleanly.", "In this post, we focus on algebraic expressions and linear equations where cross-multiplication aids step-by-step simplification — not just in fractional form, but algebraically.", "---", "### Solving the Equation: 2(5x – 4) = 7x + 2", "Let’s take a classic equation:", "[\n2(5x - 4) = 7x + 2\n]", "Step 1: Expand both sides\nDistribute the 2 across the parentheses:\n[\n2 \cdot 5x - 2 \cdot 4 = 7x + 2 \implies 10x - 8 = 7x + 2\n]", "Step 2: Move variable terms to one side\nSubtract (7x) from both sides to collect like terms:\n[\n10x - 7x - 8 = 2 \implies 3x - 8 = 2\n]", "Step 3: Isolate the variable\nAdd 8 to both sides:\n[\n3x = 10\n]", "Step 4: Solve for (x)\nDivide both sides by 3:\n[\nx = \frac{10}{3}\n]", "---", "### The Role of Cross-Multiplying in Equation Solving", "While our example doesn’t feature fractions or proportions directly, the principle of cross-multiplying is closely tied to balancing both sides of an equation. Imagine if the equation involved fractions like:", "[\n\frac{2x - 4}{5} = \frac{7x + 2}{3}\n]", "In that case, cross-multiplying — multiplying both numerator and denominator sides by the least common denominator (15 here) — is ideal:", "[\n15 \cdot \frac{2x - 4}{5} = 15 \cdot \frac{7x + 2}{3} \implies 3(2x - 4) = 5(7x + 2)\n]", "This transforms the equation into a solvable linear form, similar to the method we used in Step 1—but drastically streamlines solving by clearing denominators entirely.", "---", "### Why Cross-Multiplying Helps You Master Equations", "- Helps manage complexity: Breaking expressions into clearer parts makes solving easier.\n- Promotes accurate balancing: Ensures both sides remain equal throughout manipulation.\n- Prepares for advanced math: Builds foundational skills for working with rational expressions and proportions.\n- Enhances problem-solving flexibility: Knowing when and how to apply cross-multiplying expands your algebraic toolkit.", "---", "### Final Answer", "From our example:", "[\n2(5x - 4) = 7x + 2 \implies x = \frac{10}{3}\n]", "Cross-multiplying is more than just a shortcut — it’s a powerful concept that strengthens your ability to solve equations confidently and precisely. Practice simplifying expressions, balancing both sides, and always think strategically about how terms relate to each other.", "By mastering these steps, you’ll not only solve equations faster but also gain deeper insight into the logic behind algebra.", "---", "Key Takeaways:\n- Cross-multiplying applies beyond fractions — it’s about maintaining equality during simplification.\n- Expanding, collecting terms, and isolating variables are crucial steps.\n- Patience and practice turn algebraic manipulation into second nature.\n- Understanding cross-multiplying prepares you for rational equations and proportions.", "Start practicing with equations like 2(3x + 1) = 5x – 4, and soon cross-multiplying will feel intuitive!", "---", "Keywords: Cross-multiplying in algebra, solving linear equations, how to solve 2(5x–4) = 7x + 2, step-by-step equation solving, algebraic methods for x, simplified equation techniques.", "---", "Meta Description:\nLearn how cross-multiplying simplifies solving equations like (2(5x - 4) = 7x + 2). Step-by-step guide with full solution showing (x = \frac{10}{3}) — clear, accurate, and perfect for algebra students."]