Solution: Set $ -3x + 18 = 4x - 2 $. Solve: $ 18 + 2 = 7x \Rightarrow x = rac{20}{7} $. Substitute into $ y = -3 \cdot rac{20}{7} + 18 = -rac{60}{7} + rac{126}{7} = rac{66}{7} $. Intersection at $ \left( rac{20}{7}, rac{66}{7}

["Solving the Linear Equation: Finding the Intersection Point Step by Step", "Finding the intersection of two linear equations is a fundamental skill in algebra that appears in many areas of mathematics, science, and real-world modeling. One common problem involves solving for ( x ) in the equation ( -3x + 18 = 4x - 2 ), then using that solution to find the corresponding ( y )-value. This step-by-step guide explains how to solve this equation and determine the exact intersection point ( \left( \frac{20}{7}, \frac{66}{7} \right) ).", "### Step 1: Set Up the Equation\nStart with the equation:\n[\n-3x + 18 = 4x - 2\n]\nOur goal is to isolate ( x ) by collecting all variable terms on one side and constant terms on the other.", "### Step 2: Move All ( x )-Terms to One Side\nAdd ( 3x ) to both sides to eliminate ( -3x ) from the left-hand side:\n[\n18 = 4x + 3x - 2 \quad \Rightarrow \quad 18 = 7x - 2\n]", "### Step 3: Isolate the Constant Term\nAdd 2 to both sides to move the constant:\n[\n18 + 2 = 7x \quad \Rightarrow \quad 20 = 7x\n]", "### Step 4: Solve for ( x )\nDivide both sides by 7:\n[\nx = \frac{20}{7}\n]", "### Step 5: Substitute ( x ) into the Second Equation to Find ( y )\nUse the original equation ( y = -3x + 18 ) (or equivalently ( y = 4x - 2 )) and substitute ( x = \frac{20}{7} ).\nUsing ( y = -3 \cdot \frac{20}{7} + 18 ):\n[\ny = -\frac{60}{7} + 18 = -\frac{60}{7} + \frac{126}{7} = \frac{66}{7}\n]", "### Step 6: Final Intersection Point\nThe solution represents the coordinates where the two lines intersect:\n[\n\left( \frac{20}{7}, \frac{66}{7} \right)\n]", "---", "### Why This Matters\nSolving equations like ( -3x + 18 = 4x - 2 ) demonstrates how to find the point where two linear relationships meet. This method applies to physics, economics, engineering, and data analysis, making it a vital algebraic tool. Understanding how to isolate variables, combine like terms, and substitute solutions prepares learners for more advanced math and real-world problem solving.", "Whether you're modeling motion, optimizing costs, or analyzing trends, mastering this technique opens doors to clearer, analytical thinking.", "Key Takeaway:\nBy breaking down the equation and substituting step by step, we find the precise solution ( x = \frac{20}{7} ), leading to the intersection point ( \left( \frac{20}{7}, \frac{66}{7} \right) )."]