Then solving \( x - y = 4 \), \( x + y = 10 \), add: \( 2x = 14 \Rightarrow x = 7 \), \( y = 3 \)

["# Solving the System of Equations: ( x - y = 4 ), ( x + y = 10 )\nStep-by-Step Guide to Finding ( x = 7 ) and ( y = 3 )", "Mathematics offers powerful tools to solve real-world problems, and one of the most essential types of problems is solving systems of linear equations. This article walks you through solving the classic system:", "[\n\begin{cases} \nx - y = 4 \quad &\ ext{(Equation 1)} \\nx + y = 10 \quad &\ ext{(Equation 2)} \n\end{cases}\n]", "Understanding how to solve such equations not only helps in academics but also supports logical thinking used in engineering, economics, and data science.", "## Step-by-Step Solution", "### Step 1: Eliminate one variable by adding equations\nWe are given:\n[\nx - y = 4\n]\n[\nx + y = 10\n]", "Adding both equations eliminates ( y ):\n[\n(x - y) + (x + y) = 4 + 10\n]\n[\nx - y + x + y = 14\n]\n[\n2x = 14\n]", "### Step 2: Solve for ( x )\nDivide both sides by 2:\n[\nx = \frac{14}{2} = 7\n]\nSo, ( x = 7 )", "### Step 3: Substitute ( x = 7 ) into one equation to find ( y )\nUse Equation 2 (( x + y = 10 )):\n[\n7 + y = 10\n]\nSubtract 7 from both sides:\n[\ny = 10 - 7 = 3\n]\nThus, ( y = 3 )", "## Verification\nPlug ( x = 7 ) and ( y = 3 ) back into both original equations to confirm correctness:", "Equation 1:\n[\n7 - 3 = 4 \quad \ ext{✓ True}\n]", "Equation 2:\n[\n7 + 3 = 10 \quad \ ext{✓ True}\n]", "### How This Relates to Simple Algebraic Shortcut\nAn efficient shortcut involves adding the equations directly:\n[\nx - y = 4\n]\n[\nx + y = 10\n]\nAdd them:\n[\n2x = 14 \Rightarrow x = 7\n]\nThen substitute into ( x + y = 10 ) to get ( y = 3 )", "## Why This Method Works\nBy combining the two equations, we eliminate the variable ( y ), reducing the system to a single equation in ( x ). This algebraic elimination method is reliable and faster, especially useful in automated computation and complex systems.", "---", "Conclusion\nSolving ( x - y = 4 ) and ( x + y = 10 ) leads to ( x = 7 ) and ( y = 3 ). Whether using substitution or elimination, these foundational techniques form the basis for solving linear systems in mathematics, science, and engineering. Next time you encounter such equations, try both methods—adding equations offers a quick path to the solution!", "Keywords: solve equations, linear system, algebra, elimination method, substitution method, ( x = 7 ), ( y = 3 ), solving equations step-by-step, math tutorial, vocabulary: solve ( x - y = 4 ), solve ( x + y = 10 )", "---", "Adding to your knowledge: For further practice, try solving systems involving inequalities or more complex equations—mastering these skills unlocks deeper understanding of mathematics and its applications."]