["# Solve for x: 3(x - 4) + 2x = 5x + 6", "Learning how to solve linear equations is one of the most essential math skills you’ll use throughout your academic journey. Whether you’re tackling algebra for the first time or sharpening your problem-solving abilities, understanding how to isolate variables like x gives you the foundation to master more complex math. In this article, we’ll break down how to solve the equation:", "3(x - 4) + 2x = 5x + 6", "---", "## Step-by-Step Solution", "### 1. Expand the expression
\nStart by distributing the 3 across the parentheses:
\n[ 3(x - 4) + 2x = 3x - 12 + 2x ]", "Combine like terms on the left-hand side:
\n[ 3x + 2x - 12 = 5x - 12 ]
\nSo the equation becomes:
\n[ 5x - 12 = 5x + 6 ]", "### 2. Subtract 5x from both sides
\nTo isolate x, subtract 5x from both sides:
\n[ 5x - 12 - 5x = 5x + 6 - 5x ]
\nSimplify:
\n[ -12 = 6 ]", "### 3. Interpret the result
\nWhat does this mean? The final statement, -12 = 6, is false. This indicates that there is no solution to the equation — the two sides never meet, meaning no value of x satisfies the equation.", "---", "## Why Does This Happen?", "When solving equations like 3(x – 4) + 2x = 5x + 6, it’s common to reach a contradiction — a statement that’s always false, such as 5 = 6. This signifies that the original equation is inconsistent, meaning it has no solution. Graphically, this corresponds to two lines that never intersect (parallel lines with different y-intercepts), despite having the same slope.", "---", "## How to Confirm No Solution", "### Algebraically:
\nAfter simplifying to –12 = 6, no value of x resolves the contradiction.", "### Graphically:
\nPlot both sides as linear functions:
\nLeft side: ( y = 5x - 12 )
\nRight side: ( y = 5x + 6 )
\nBoth lines have a slope of 5 but different y-intercepts (–12 vs. 6), so they are parallel and never cross.", "---", "## Practice Problem", "Try solving:
\n3(x – 4) + 2x = 5x + 6
\nYou’ll find the equation simplifies to –12 = 6 — no solution.", "---", "## Key Takeaways", "- Expand expressions carefully using the distributive property.
\n- Combine like terms to simplify both sides.
\n- Watch for contradictions — if they occur, the equation has no solution.
\n- Graphical interpretation helps visualize why no solution exists: parallel lines.", "Mastering these steps builds stronger problem-solving skills and prepares you for more advanced algebra, calculus, and real-world applications where equations model systems and relationships.", "---", "## Summary Quick Answer", "To solve 3(x – 4) + 2x = 5x + 6, expand, combine terms, and simplify to:
\n[ 5x - 12 = 5x + 6 ]
\nSubtracting 5x from both sides yields:
\n[ -12 = 6 ] — a contradiction.
\nTherefore, the equation has no solution.", "---", "If you’re learning algebra or brushing up on equations, remember: recognizing when an equation has no solution is just as important as finding the solution itself. Keep practicing, stay curious, and keep solving!", "---", "Keywords: solve for x, linear equations, algebraic equations, no solution, solve 3(x–4) + 2x = 5x + 6, step-by-step algebra, contradiction in equations, solving by expanding, parallel lines algebraically, math problem solving tips", "Meta Description: Learn how to solve the equation 3(x – 4) + 2x = 5x + 6. Discover step-by-step instructions, why this equation has no solution, and tips for mastering linear equations. Perfect for students and beginner math learners."]
