Solve the system of equations: \( 2x + y = 10 \) and \( x - y = 2 \).
["# Solve the System of Equations: ( 2x + y = 10 ) and ( x - y = 2 )", "Solving systems of linear equations is a foundational skill in algebra, essential for students, educators, and professionals in STEM fields. This article provides a clear, step-by-step guide to solving a key example: Solve the system of equations: ( 2x + y = 10 ) and ( x - y = 2 ). By mastering this method, you’ll build the confidence and technique needed to tackle more complex problems.", "## Why Solve Systems of Equations?", "Systems of equations arise in many real-world situations—from economics and engineering to computer programming and data analysis. By finding the intersection point of two equations, we determine the unique values of variables that satisfy both conditions simultaneously.", "In this case, the system:", "[\n\begin{cases}\n2x + y = 10 \quad \ ext{(Equation 1)} \\nx - y = 2 \quad \ ext{(Equation 2)}\n\end{cases}\n]", "is designed to help you practice elimination and substitution—two core methods for solving equations.", "## Step-by-Step Solution", "### Method 1: Elimination (Addition Method)", "The elimination method works by manipulating the equations to eliminate one variable, making it easy to solve for the other.", "Step 1: Align the equations", "[\n\begin{align}\n2x + y &= 10 \quad \ ext{(1)} \\n\quad x - y &= 2 \quad \ ext{(2)}\n\end{align}\n]", "Notice that the ( y )-terms have opposite signs—this is ideal for elimination.", "Step 2: Add the equations to eliminate ( y )", "Add Equation 1 and Equation 2 directly:", "[\n(2x + y) + (x - y) = 10 + 2\n]", "Simplify:", "[\n2x + x + y - y = 12 \Rightarrow 3x = 12\n]", "Step 3: Solve for ( x )", "[\nx = \frac{12}{3} = 4\n]", "Step 4: Substitute ( x = 4 ) into one of the original equations", "Use Equation 2: ( x - y = 2 )", "[\n4 - y = 2\n]", "Solve for ( y ):", "[\ny = 4 - 2 = 2\n]", "### Method 2: Substitution (Alternative Check)", "For verification, use substitution. From Equation 2:", "[\nx - y = 2 \Rightarrow x = y + 2\n]", "Substitute ( x = y + 2 ) into Equation 1:", "[\n2(y + 2) + y = 10\n]", "Expand and simplify:", "[\n2y + 4 + y = 10 \Rightarrow 3y + 4 = 10\n]", "Solve:", "[\n3y = 6 \Rightarrow y = 2\n]", "Back-substitute:", "[\nx = y + 2 = 2 + 2 = 4\n]", "Both methods yield the same solution.", "## Final Answer", "The solution to the system is:", "[\n\boxed{x = 4,\ y = 2}\n]", "The ordered pair ( (4, 2) ) satisfies both equations:", "- ( 2(4) + 2 = 8 + 2 = 10 )\n- ( 4 - 2 = 2 )", "---", "## Why This Example Matters", "This system demonstrates how two linear equations intersect at a single point—unique solutions in geometry and algebra. It’s a typical setup for problems involving budgets, rates, or balance points. Understanding how to solve it enhances fluency with algebraic techniques.", "---", "## Tips to Master Solving Systems of Equations", "- Check your work: Plug the values of ( x ) and ( y ) back into both equations to confirm they hold true.\n- Use elimination when coefficients are opposites or can be adjusted easily.\n- Substitution works best when one variable is already solved or easily isolated.\n- Practice with different numbers to build speed and accuracy.", "---", "## Explore More", "Need help with more complex systems or word problems? Explore our full library of guided tutorials on linear equations, elimination methods, and real-world applications using simultaneous equations.", "Start solving systems today—and unlock the power of algebra!", "---", "Keywords: solve system of equations, linear equations, elimination method, substitution method, algebra tutorials, solving equations, unique solution, ( 2x + y = 10 ), ( x - y = 2 ), step-by-step math"]
