Solve the system: \( 2x + y = 10 \) and \( x - y = 2 \).
["Solving the System of Equations: ( 2x + y = 10 ) and ( x - y = 2 )", "Solving systems of linear equations is a fundamental skill in algebra, widely applied in science, engineering, economics, and everyday problem-solving. One classic example is the system:", "[\n\begin{cases}\n2x + y = 10 \\nx - y = 2\n\end{cases}\n]", "This article guides you step-by-step through solving this system using the substitution and elimination methods, showcasing efficient techniques to find the values of (x) and (y) that satisfy both equations.", "---", "### Why Solve Linear Systems?", "Linear systems model relationships between variables in real-world scenarios—such as budgeting, mixing solutions, or balancing equations. Mastering these methods equips you with tools to tackle more complex mathematical and practical problems.", "---", "### Method 1: Solving by Substitution", "The substitution method involves solving one equation for one variable and substituting that expression into the other equation.", "Step 1: Solve the second equation for (x):", "From\n[\nx - y = 2\n]\nadd (y) to both sides:\n[\nx = y + 2\n]", "Step 2: Substitute into the first equation:", "Replace (x) in (2x + y = 10) with (y + 2):\n[\n2(y + 2) + y = 10\n]", "Step 3: Simplify and solve for (y):", "[\n2y + 4 + y = 10 \\n3y + 4 = 10 \\n3y = 6 \\ny = 2\n]", "Step 4: Back-substitute to find (x):", "Using (x = y + 2):\n[\nx = 2 + 2 = 4\n]", "Solution:\n[\nx = 4, \quad y = 2\n]", "---", "### Method 2: Solving by Elimination (Addition)", "The elimination method involves adding or subtracting equations to eliminate one variable.", "Step 1: Write both equations:", "[\n2x + y = 10 \quad \ ext{(Equation 1)} \\nx - y = 2 \quad \ ext{(Equation 2)}\n]", "Notice that the (y) terms have opposite signs ((+y) and (-y)), making elimination straightforward.", "Step 2: Add both equations to eliminate (y):", "[\n(2x + y) + (x - y) = 10 + 2 \\n2x + x + y - y = 12 \\n3x = 12\n]", "Step 3: Solve for (x):", "[\nx = 4\n]", "Step 4: Substitute back to find (y):", "Use (x - y = 2):\n[\n4 - y = 2 \Rightarrow y = 2\n]", "---", "### Final Answer", "The solution to the system is:\n[\n\boxed{x = 4,\ y = 2}\n]", "---", "### Verification", "Substitute (x = 4) and (y = 2) back into both original equations to confirm:", "1. (2(4) + 2 = 8 + 2 = 10) ✅\n2. (4 - 2 = 2) ✅", "Both equations are satisfied, confirming the solution is correct.", "---", "### Summary", "Solving the system (2x + y = 10) and (x - y = 2) illustrates two powerful algebraic techniques: substitution and elimination. Whether you prefer isolating variables or manipulating equations to eliminate terms, both methods yield the accurate solution efficiently. Practice these strategies to build a strong foundation for solving more complex systems in advanced mathematics.", "---", "Keywords: solve system of equations, linear equations, substitution method, elimination method, solve (2x + y = 10) and (x - y = 2), algebra technique, linear algebra solutions, step-by-step system solving, chemistry and physics applications of systems, educational math tutorial."]
