How To Find Intersection Of 2 Lines

Intersection of two directly lines (Coordinate Geometry)

The betoken of intersection of two non-parallel lines can be found from the
equations of the 2 lines.

Try this Elevate whatsoever of the 4 points below to move the lines. Note where they intersect.

To find the intersection of two straight lines:

Commencement nosotros demand the equations of the two lines. If yous do not accept the equations, see Equation of a line - gradient/intercept form and Equation of a line - bespeak/slope course (If 1 of the lines is vertical, see the department below).
So, since at the point of intersection, the ii 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.

This gives us the 10 and y coordinates of the intersection.

Case

So for example, if we have ii lines that accept the following equations (in slope-intercept course):

y = 3x-3

y = 2.3x+4

At the bespeak of intersection they volition both accept the same y-coordinate value, so nosotros set the equations equal to each other:

3x-3 = 2.3x+4

This gives us an equation in one unknown (10) which we can solve: Re-arrange to get 10 terms on left

3x - 2.3x = 4+3

Combining like terms

0.7x = vii

Giving

x = 10

To observe y, simply fix x equal to 10 in the equation of either line and solve for y: Equation for a line (Either line will do)

y = 3x - three

Set ten equal to 10

y = 30 - three

Giving

y = 27

Nosotros now have both x and y, so the intersection signal is (10, 27)

Which equation course to utilize?

Retrieve that lines can be described by the slope/intercept form and bespeak/slope form of the equation. Finding the intersection works the same way for both. Merely gear up the equations equal as above. For example, if you lot had two equations in betoken-slope form:

y = 3(x-3) + ix

y = 2.one(x+2) - 4

just set up them equal:

3(x-3) + 9 = 2.1(x+ii) - 4

and go on every bit above, solving for 10, then substituting that value into either equation to find y.

The two equations demand non even be in the same grade. Simply prepare them equal to each other and proceed in the usual style.

When one line is vertical

When one of the lines is vertical, it has no defined slope, so its equation will look something similar ten=12. See Vertical lines (Coordinate Geometry). Nosotros find the intersection slightly differently. Suppose we have the lines whose equations are

y = 3x-iii	A line sloping up and to the correct
ten = 12	A vertical line

On the vertical line, all points on it have an 10-coordinate of 12 (the definition of a vertical line), so we simply set x equal to 12 in the showtime equation and solve information technology for y.
Equation for a line:

y = 3x - 3

Ready x equal to 12 Using the equation of the second (vertical) line

y = 36 - iii

Giving

y = 33

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

If both lines are vertical, they are parallel and have no intersection (see below).

When they are parallel

When two lines are parallel, they do not intersect anywhere. If you endeavour to observe the intersection, the equations will be an absurdity. For instance the lines y=3x+iv and y=3x+8 are parallel because their slopes (3) are equal. See Parallel Lines (Coordinate Geometry). If you try the above process you lot would write 3x+4 = 3x+8. An obvious impossibility.

Segments and rays might not intersect at all

Fig i. Segments do non intersect

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

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

Things to try

In the above diagram, printing '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 whatever new location where the intersection is all the same visible. Calculate the slopes of the lines and the point of intersection. Click 'show details' to verify your result.

Limitations

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