# Point of intersection of two lines

• dcramps
In summary, to find the point of intersection of two lines, you can set two of the equations equal to each other and solve for the parameters. Then, plug the values of the parameters into the third equation to confirm that they are equal. Finally, plug the values into all of the original equations to find the intersection point.

## Homework Statement

Find the point of intersection of two lines
x = -9 + 5t
y = 1 + t
z = 10 - 4t

and

x = -2 -3t
y = 5 + 2t
z = 5 + 3t

N/A

## The Attempt at a Solution

I have read that you should set two of the equations equal to find the value of t, and that value of t should hold for the third equation, but whenever I do that I get two different values for t, and neither of them hold up in the third equation. What am I doing wrong?

Hi dcramps!

Where the two lines intersect, the parameter t will be different on each line.

So call the parameter for the second line u, then set two of the equations equal to find the values of t and u.

Thanks! Now I have my t and u values, and plugging them into the third equation gives me equality...but where do I go from here? Do I plug them into all of the original equations and use the results as my intersection point?

Yup!

Excellent. Thanks for your help :)

## 1. What is the point of intersection of two lines?

The point of intersection of two lines is the single point where the two lines meet or cross each other. It is the common solution to the equations of the two lines, and it represents the coordinates of the point where the two lines intersect on a graph.

## 2. How do you find the point of intersection of two lines?

To find the point of intersection of two lines, you need to solve the equations of the two lines simultaneously. This can be done by using the substitution or elimination method. Once you have solved for the values of x and y, you can plug them into the equation of either line to find the coordinates of the point of intersection.

## 3. Can two lines intersect at more than one point?

No, two lines in a plane can only intersect at one point. This is because two distinct lines can only have one common solution when their equations are solved simultaneously. If two lines have the same equation, they are considered to be the same line and can intersect at infinitely many points.

## 4. What does it mean if two lines do not intersect?

If two lines do not intersect, it means that they are parallel to each other. This means that they have the same slope and will never meet or cross each other. In other words, there is no point of intersection between two parallel lines.

## 5. Can two lines intersect at a right angle?

Yes, two lines can intersect at a right angle. This happens when the two lines are perpendicular to each other, meaning that they form a 90 degree angle where they meet. In this case, the product of their slopes is -1, and the lines can be represented by the equations y = mx + b and y = (-1/m)x + b, where m is the slope of the first line.