Question: Find the intersection point of the lines $ 2x + y = 10 $ and $ x - y = 2 $.
["# Find the Intersection Point of the Lines $ 2x + y = 10 $ and $ x - y = 2 $", "Determining the intersection point of two lines is a fundamental skill in algebra, useful in geometry, engineering, and real-world problem solving. In this article, we’ll learn how to find the exact point where the lines defined by the equations $ 2x + y = 10 $ and $ x - y = 2 $ meet.", "## Step 1: Understanding the Problem", "The intersection point of two lines is the single point $(x, y)$ that satisfies both equations simultaneously. By solving the system of linear equations:", "$$\n\begin{align}\n(1)\quad & 2x + y = 10 \\n(2)\quad & x - y = 2 \\n\end{align}\n$$", "we can find the values of $x$ and $y$ that satisfy both simultaneously.", "## Step 2: Choose a Method to Solve the System", "There are two common methods: substitution and elimination. Here, we’ll use substitution because the second equation is simple and easily solved for one variable.", "From Equation (2):\n$$\nx - y = 2 \implies x = y + 2\n$$", "## Step 3: Substitute into the First Equation", "Substitute $ x = y + 2 $ into Equation (1):", "$$\n2(y + 2) + y = 10\n$$", "Simplify:", "$$\n2y + 4 + y = 10 \implies 3y + 4 = 10\n$$", "## Step 4: Solve for $ y $", "Subtract 4 from both sides:", "$$\n3y = 6 \implies y = 2\n$$", "## Step 5: Find $ x $", "Now substitute $ y = 2 $ back into $ x = y + 2 $:", "$$\nx = 2 + 2 = 4\n$$", "## Step 6: State the Intersection Point", "The lines intersect at the point $(4, 2)$.", "---", "## Why This Matters", "Finding the intersection of lines is essential in modeling real-life scenarios such as determining collision points in navigation, optimizing resources using constraint lines, or plotting designs in engineering.", "### Summary:", "- The two lines intersect at $(4, 2)$.\n- By substitution using $ x = y + 2 $, we solved the system efficiently.\n- Verification: Plug $ x = 4 $, $ y = 2 $ into both equations:\n - $ 2(4) + 2 = 10 $ ✔️\n - $ 4 - 2 = 2 $ ✔️", "---", "### Final Answer:", "$$\n\boxed{(4,\ 2)}\n$$", "Use this method next time you need the intersection of two straight lines!"]
