The point of
intersection
of two
non-parallel
lines can be found from the

equations of the two lines.

equations of the two lines.

Try this
Drag any of the 4 points below to move the lines. Note where they intersect.

To find the intersection of two straight lines:

- First we need the equations of the two lines. If you do not have the equations, see Equation of a line - slope/intercept form and Equation of a line - point/slope form (If one of the lines is vertical, see the section below).
- Then, since at the point of intersection, the two equations will have the same values of x and y, we set the two equations equal to each other. This gives an equation that we can solve for x
- We substitute that x value in one of the line equations (it doesn't matter which) and solve it for y.

y = 3x-3

y = 2.3x+4

At the point of intersection they will both have the same y-coordinate value, so we set the equations equal to each other:
3x-3 = 2.3x+4

This gives us an equation in one unknown (3x - 2.3x = 4+3

Combining like terms
0.7x = 7

Giving
x = 10

To find y, simply set x equal to 10 in the equation of either line and solve for y:
Equation for a line (Either line will do)
y = 3x - 3

Set x equal to 10
y = 30 - 3

Giving
y = 27

We now have both x and y, so the intersection point is (10, 27)
y = 3(x-3) + 9

y = 2.1(x+2) - 4

simply set them equal:
3(x-3) + 9 = 2.1(x+2) - 4

and proceed as above, solving for x, then substituting that value into either equation to find y.
The two equations need not even be in the same form. Just set them equal to each other and proceed in the usual way.

y = 3x-3 | A line sloping up and to the right |

x = 12 | A vertical line |

On the vertical line, all points on it have an x-coordinate of 12 (the definition of a vertical line), so we simply set x equal to 12 in the first equation and solve it for y.

Equation for a line:

y = 3x - 3

Set x equal to 12 Using the equation of the second (vertical) line
y = 36 - 3

Giving
y = 33

So the intersection point is at (12,33).

If

Fig 1. Segments do not intersect

In the case of two non-parallel lines, the intersection will always be on the lines somewhere. But in the case of line segments or rays which have a limited length, they might not actually intersect.

In Fig 1 we see two line segments that do not overlap and so have no point of intersection. However, if you apply the method above to them, you will find the point where they would have intersected if extended enough.

- In the above diagram, press 'reset'.
- Drag any of the points A,B,C,D around and note the location of the intersection of the lines.
- Drag a point to get two parallel lines and note that they have no intersection.
- Click 'hide details' and 'show coordinates'. Move the points to any new location where the intersection is still visible. Calculate the slopes of the lines and the point of intersection. Click 'show details' to verify your result.

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.

For more see Teaching Notes

- Introduction to coordinate geometry
- The coordinate plane
- The origin of the plane
- Axis definition
- Coordinates of a point
- Distance between two points

- Introduction to Lines

in Coordinate Geometry - Line (Coordinate Geometry)
- Ray (Coordinate Geometry)
- Segment (Coordinate Geometry)
- Midpoint Theorem

- Cirumscribed rectangle (bounding box)
- Area of a triangle (formula method)
- Area of a triangle (box method)
- Centroid of a triangle
- Incenter of a triangle
- Area of a polygon
- Algorithm to find the area of a polygon
- Area of a polygon (calculator)
- Rectangle
- Square
- Trapezoid
- Parallelogram

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