To find the intersection of the lines \( y = 2x + 3 \) and \( y = -x + 7 \), we set the equations equal to each other:

["# Finding the Intersection of Two Lines: Solving ( y = 2x + 3 ) and ( y = -x + 7 )", "When working with linear equations, one of the most common tasks is determining where two lines intersect. The point of intersection represents the solution to the system of equations formed by these lines. In this article, we’ll explore how to find the intersection point of the lines given by:", "[\ny = 2x + 3\n]\nand\n[\ny = -x + 7\n]", "## Why Find the Line Intersection?", "The intersection of two lines signifies a single point where both equations are true simultaneously. Solving such systems accurately is essential in math, physics, engineering, and computer graphics—any field where relationships between linear variables matter.", "## Step-by-Step: Solving ( y = 2x + 3 ) and ( y = -x + 7 )", "To locate the intersection, we use the fundamental principle that at the point of intersection, both equations yield the same ( y )-value for a shared ( x )-value. This means we can set the right-hand sides of the two equations equal to each other:", "[\n2x + 3 = -x + 7\n]", "Now solve for ( x ):", "1. Add ( x ) to both sides:\n[\n2x + x + 3 = 7\n]\n[\n3x + 3 = 7\n]", "2. Subtract 3 from both sides:\n[\n3x = 4\n]", "3. Divide both sides by 3:\n[\nx = \frac{4}{3}\n]", "## Now substitute ( x = \frac{4}{3} ) into one of the original equations", "We’ll use ( y = 2x + 3 ) for simplicity:", "[\ny = 2\left(\frac{4}{3}\right) + 3 = \frac{8}{3} + 3 = \frac{8}{3} + \frac{9}{3} = \frac{17}{3}\n]", "## Conclusion: The Point of Intersection", "The two lines intersect at the coordinate:", "[\n\left( \frac{4}{3},\ \frac{17}{3} \right)\n]", "This means the solution to the system ( y = 2x + 3 ) and ( y = -x + 7 ) is the unique point where both equations cross—ideal for graphing, verification, or real-world modeling.", "For visual learners, plotting these lines confirms the intersection visually. For math enthusiasts and students, mastering this technique strengthens understanding of linear systems and algebraic problem-solving.", "---", "Keywords: find intersection of two lines, solve ( y = 2x + 3 ) and ( y = -x + 7 ), how to solve linear equations, intersection point, algebra tutorial, linear systems solution", "Meta description: Learn step-by-step how to find the intersection of ( y = 2x + 3 ) and ( y = -x + 7 ) by setting the equations equal and solving for ( x ) and ( y )."]