Solution: Set \( 3x + 1 = -2x + 11 \). Solve for \( x \): \( 5x = 10 \) → \( x = 2 \). Substitute into \( y = 3(2) + 1 = 7 \). The intersection is \( (2, 7) \). oxed{(2, 7)}

["How to Solve Linear Equations: A Step-by-Step Guide to Finding the Intersection Point", "Learning to solve linear equations is a fundamental skill in algebra, essential for fields like engineering, economics, and data science. One common task is finding the solution to an equation like ( 3x + 1 = -2x + 11 ), which represents finding the point where two linear expressions intersect. In this article, we’ll walk through the solution method step-by-step, explain how to isolate the variable, and verify the final intersection point — culminating in the precise solution: ( \boxed{(2, 7)} ).", "---", "### Understanding the Equation", "The equation ( 3x + 1 = -2x + 11 ) compares two linear expressions. Finding ( x ) where both sides are equal helps determine their intersection point on the coordinate plane. This process teaches not only algebraic manipulation but also prepares learners for graphing and real-world applications involving linear relationships.", "---", "### Step 1: Move All Terms Involving ( x ) to One Side", "Begin by gathering all terms containing ( x ) on one side of the equation. Add ( 2x ) to both sides to eliminate ( -2x ):", "[\n3x + 2x + 1 = 11\n]", "[\n5x + 1 = 11\n]", "Now, subtract 1 from both sides:", "[\n5x = 11 - 1\n]", "[\n5x = 10\n]", "---", "### Step 2: Solve for ( x )", "Divide both sides by 5 to isolate ( x ):", "[\nx = \frac{10}{5} = 2\n]", "This value ( x = 2 ) is where the two functions balance and intersect.", "---", "### Step 3: Find the Corresponding ( y )-Value (Optional for Two-Variable Systems)", "If this equation is part of a system (e.g., modeling two linear functions), substitute ( x = 2 ) into one of the original expressions to find ( y ). Using ( y = 3x + 1 ):", "[\ny = 3(2) + 1 = 6 + 1 = 7\n]", "Thus, the intersection point is ( (2, 7) ).", "---", "### Why This Intersection Matters", "The point ( (2, 7) ) represents the solution to the system—where ( 3x + 1 = -2x + 11 ) and where both expressions yield ( y = 7 ). This concept applies broadly in math and science: intersections reveal common solutions, balance, or equilibrium.", "---", "### Final Thoughts", "Mastering linear equation solving—like solving ( 3x + 1 = -2x + 11 )—builds a foundation for more complex math. By isolating variables, combining like terms, and verifying solutions, anyone can confidently solve for ( x ), substitute back, and confirm intersection points such as:", "[\n\boxed{(2, 7)}\n]", "Whether you’re a student, teacher, or math enthusiast, tracking these steps ensures fluency and clarity in working with linear relationships. Keep practicing—and soon, solving equations will feel intuitive!"]