Intersect — Definition, Formula & Examples
Intersect means to cross or meet at a point. Two lines intersect when they share exactly one point in common, and that shared point is called the point of intersection.
Two geometric figures intersect if their sets of points have a non-empty overlap. For two distinct, non-parallel lines in a plane, the intersection is a unique point whose coordinates satisfy both equations simultaneously.
How It Works
To find where two lines intersect on the coordinate plane, you solve their equations at the same time (a system of equations). The solution gives you the - and -values of the point where the lines cross. If the lines are parallel, they never meet, so there is no intersection. If the lines are identical, every point is shared, so they intersect everywhere.
Worked Example
Problem: Find the point where the lines y = 2x + 1 and y = -x + 7 intersect.
Set the equations equal: Since both expressions equal y, set them equal to each other.
Solve for x: Add x to both sides, then subtract 1 from both sides.
Solve for y: Substitute x = 2 into either original equation.
Answer: The two lines intersect at the point .
Why It Matters
Finding intersections is central to solving systems of equations in algebra and is used constantly in coordinate geometry. In real life, GPS navigation, computer graphics, and engineering all rely on calculating where lines, curves, or surfaces meet.
Common Mistakes
Mistake: Assuming two lines always intersect.
Correction: Parallel lines have the same slope but different y-intercepts, so they never cross. Always check whether the slopes are equal before solving.