Determine the point of intersection of two lines

The intersection of two lines is the ordered pair that satisfies both linear equations, found by equating expressions, solving a simultaneous linear system, or reading the crossing point from a coordinate graph. The reasoning distinguishes one intersection for nonparallel lines, no intersection for distinct parallel lines, and infinitely many points for coincident lines; the result can be verified by substitution into both equations. The focus is two-dimensional coordinate geometry, excluding skew lines and more advanced vector or parametric treatments.

Detailed Explanation: Determine the point of intersection of two lines

The point of intersection is the ordered pair $(x,y)$ that lies on both lines. To find it, set the two expressions for $y$ equal to each other and solve.

Example

Find the point of intersection of

y=2x+1

and

y=-x+7.

Since both equations equal $y$ , equate their right-hand sides:

2x+1=-x+7

Solve for $x$ :

3x=6

x=2

Substitute $x=2$ into either original equation:

y=2(2)+1=5

Therefore, the lines intersect at

\boxed{(2,5)}.

Check

Substitute $(2,5)$ into both equations:

5=2(2)+1

and

5=-2+7.

Both equations are true, so $(2,5)$ is the correct intersection point.

For two lines, there are three possibilities:

One intersection: the lines are not parallel.
No intersection: the lines are distinct and parallel.
Infinitely many intersections: the two equations describe the same line.

Learn by doing: Determine the point of intersection of two lines

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Linear Equations - Find Intersection (Decimal) - Two Linear Equations

Earned ?