حل نظام المعادلات: \(2x + 3y = 12\) و \(x - y = 2\).

["How to Solve the System of Equations: (2x + 3y = 12) and (x - y = 2)", "Solving systems of linear equations is a fundamental skill in algebra and a cornerstone for understanding more advanced math topics. One commonly taught and practically useful system is the set:", "[\n\begin{cases}\n2x + 3y = 12 \\nx - y = 2\n\end{cases}\n]", "In this article, we will walk through step-by-step methods to solve this system and explore effective techniques to build confidence in finding exact solutions efficiently.", "---", "### Why Solve Systems of Equations?", "Systems of equations model real-world problems where multiple unknowns interact, such as economics, engineering, physics, and computer science. Mastering their solution helps analyze relationships between variables and find optimal points, intersections, or balances.", "---", "### Step-by-Step Method: Substitution", "Let’s solve the system using substitution, which is straightforward when one equation isolates a variable.", "Given system:\n(1) (2x + 3y = 12)\n(2) (x - y = 2)", "Step 1: Solve equation (2) for (x) or (y)\nFrom equation (2), solve for (x):\n[\nx = y + 2\n]", "Step 2: Substitute (x = y + 2) into equation (1)\nReplace (x) in equation (1):\n[\n2(y + 2) + 3y = 12\n]\nExpand:\n[\n2y + 4 + 3y = 12\n]\nCombine like terms:\n[\n5y + 4 = 12\n]\nSubtract 4 from both sides:\n[\n5y = 8\n]\nDivide by 5:\n[\ny = \frac{8}{5} = 1.6\n]", "Step 3: Substitute (y = \frac{8}{5}) back into (x = y + 2)\n[\nx = \frac{8}{5} + 2 = \frac{8}{5} + \frac{10}{5} = \frac{18}{5} = 3.6\n]", "---", "### Final Solution\n[\n\boxed{x = \frac{18}{5}, \quad y = \frac{8}{5}}\n]", "---", "### Alternative Method: Elimination", "To reinforce understanding, we can also solve by elimination.", "Multiply equation (2) by 3 to align (y) coefficients:\n[\n3(x - y) = 3 \cdot 2 \Rightarrow 3x - 3y = 6\n]", "Now add to equation (1):\n[\n(2x + 3y) + (3x - 3y) = 12 + 6 \Rightarrow 5x = 18\n]\n[\nx = \frac{18}{5}\n]", "Substitute back into (x - y = 2):\n[\n\frac{18}{5} - y = 2 \Rightarrow y = \frac{18}{5} - 2 = \frac{8}{5}\n]", "Same result — confirming consistency and accuracy.", "---", "### Verifying the Solution", "Plug (x = \frac{18}{5}, y = \frac{8}{5}) into both equations:", "Equation (1):\n[\n2\left(\frac{18}{5}\right) + 3\left(\frac{8}{5}\right) = \frac{36}{5} + \frac{24}{5} = \frac{60}{5} = 12 \quad \ ext{✓}\n]", "Equation (2):\n[\n\frac{18}{5} - \frac{8}{5} = \frac{10}{5} = 2 \quad \ ext{✓}\n]", "Both equations satisfied — solution is correct.", "---", "### Tips for Efficient Solving", "- Choose the method that simplifies your variables. If one equation already isolates a variable, substitution is faster.\n- Check each step to avoid arithmetic errors, especially with fractions.\n- Use elimination when coefficients align nicely or can be easily matched.\n- Practice helps recognize patterns and speeds up problem-solving.", "---", "### Conclusion", "Solving the system (2x + 3y = 12) and (x - y = 2) yields the unique solution:\n[\nx = \frac{18}{5}, \quad y = \frac{8}{5}\n]", "Mastering these techniques not only secures a solid algebra foundation but empowers students to tackle real-world modeling challenges confidently. Whether in academics or applied fields, understanding how to solve systems of equations is essential and rewarding.", "---", "Keywords:\nSolve linear equations, system of equations, substitution method, elimination method, solve (2x + 3y = 12) and (x - y = 2), algebra practice, linear algebra basics", "Meta Description:\nLearn how to solve the system of equations (2x + 3y = 12) and (x - y = 2) step-by-step using substitution and elimination. Find accurate, verified solutions with explanations and tips for mastering systems of equations."]