To find the intersection point of the lines \(y = 2x + 3\) and \(y = -x + 5\), we set the equations equal to each other:

["# How to Find the Intersection Point of Two Lines: A Step-by-Step Guide with (y = 2x + 3) and (y = -x + 5)", "Finding the intersection point of two lines is a fundamental skill in algebra and coordinate geometry. It allows students, educators, and math enthusiasts to visualize where two linear equations meet. In this article, we’ll walk through the process of determining where the lines (y = 2x + 3) and (y = -x + 5) intersect using the key method: setting the equations equal to each other.", "## Why Find the Intersection Point?", "The intersection point of two lines represents the solution to a system of linear equations. It tells us the values of (x) and (y) that satisfy both equations simultaneously. This has practical applications in fields like engineering, economics, and data analysis where overlapping solutions indicate key relationships.", "---", "## The Equations We Are Solving", "We want to find the point ((x, y)) that lies on both lines:", "[\ny = 2x + 3 \quad \ ext{(Equation 1)}\n]\n[\ny = -x + 5 \quad \ ext{(Equation 2)}\n]", "Since both right-hand sides equal (y), we can set the right-hand sides equal to each other:", "[\n2x + 3 = -x + 5\n]", "This step eliminates (y) and allows us to solve for (x).", "---", "## Step 1: Solve for (x)", "To solve (2x + 3 = -x + 5), start by collecting like terms. Add (x) to both sides:", "[\n2x + x + 3 = 5\n]\n[\n3x + 3 = 5\n]", "Next, subtract 3 from both sides:", "[\n3x = 2\n]", "Now divide both sides by 3:", "[\nx = \frac{2}{3}\n]", "---", "## Step 2: Substitute (x) back to find (y)", "Now that we have (x = \frac{2}{3}), substitute this value into either original equation to find (y). We’ll use Equation 1:", "[\ny = 2x + 3 = 2\left(\frac{2}{3}\right) + 3 = \frac{4}{3} + 3\n]", "Convert 3 to thirds: (3 = \frac{9}{3}), so", "[\ny = \frac{4}{3} + \frac{9}{3} = \frac{13}{3}\n]", "---", "## The Intersection Point", "The coordinates of the intersection are:", "[\n\left( \frac{2}{3},\ \frac{13}{3} \right)\n]", "This means the two lines cross at the point (\left(\frac{2}{3}, \frac{13}{3}\right)) in the coordinate plane.", "---", "## Verification", "To ensure accuracy, substitute (x = \frac{2}{3}) and (y = \frac{13}{3}) into both original equations:", "- Equation 1:\n (y = 2x + 3)\n (\frac{13}{3} = 2\left(\frac{2}{3}\right) + 3 = \frac{4}{3} + \frac{9}{3} = \frac{13}{3}) ✓", "- Equation 2:\n (y = -x + 5)\n (\frac{13}{3} = -\frac{2}{3} + 5 = -\frac{2}{3} + \frac{15}{3} = \frac{13}{3}) ✓", "Both equations confirm the solution is correct.", "---", "## Mathematical Insight: Why This Works", "Setting the two expressions for (y) equal is valid when both equations describe (y) as a function of (x). The resulting algebraic equation represents a single linear equation in one variable, allowing a unique, precise solution. This method works only when the system is linear and consistent—meaning the lines either cross at exactly one point or are parallel (no intersection).", "---", "## Conclusion", "Finding the intersection point of two lines like (y = 2x + 3) and (y = -x + 5) is straightforward when you equate the expressions and solve algebraically. By setting (2x + 3 = -x + 5), solving for (x), and substituting back, we determined the intersection at (\left(\frac{2}{3}, \frac{13}{3}\right)). This technique forms the foundation for solving systems of equations in algebra and higher mathematics.", "Whether you're solving for geometry, modeling real-world problems, or preparing for advanced math courses, mastering this intersection method is essential.", "---", "Keywords: intersection point of lines, solve linear equations, system of equations algebra, find intersection algebraically, line equations, (y = mx + b\ intersection, coordinate geometry, algebra tutorial", "Meta Description: Learn how to find the intersection point of the lines (y = 2x + 3) and (y = -x + 5) by setting the equations equal, solving for (x), and substituting back to find (y). Step-by-step guide with verification."]