# Equations of Lines/Multivariable Calculus

• EngnrMatt
In summary, the given lines intersect at a point and to determine the point of intersection, one can solve for t and s using the given equations and then substitute either t in L1 or s in L2 to find the coordinates of the point of intersection. It is important to double check the calculations to ensure accuracy.
EngnrMatt

## Homework Statement

Determine whether the lines
L1:x=t, y=16+4t ,z=8+t
and
L2:x=−7+2t, y=−8+6t, z=−3+4t
intersect, are skew, or are parallel. If they intersect, determine the point of intersection

## Homework Equations

t = -7 + 2s

16 + 4t = -8 + 6s

8 + t = -3 + 4s

## The Attempt at a Solution

I solved the first two equations for s and t then plugged them into the third which confirmed that the lines intersect. To determine the point of intersection, I figured that setting the parametric equations equal to each other and solving for t would give the correct answer, but that doesn't seem to be the right way (I got 7 for the x-coordinate of the intersection).

So my question is how do I find the intersection?

Once you found out that the lines interesected by solving for t and s, substituting either t in L1 or s in L2 would give you the point of intersection. They should match up if you do both. (And I'm not sure about 7 for the x-coordinate, double check your math).

Yep, you're right, thanks! It worked.

Sure thing!

## 1. What is the equation of a line?

The equation of a line is represented in the form y = mx + b, where m is the slope of the line and b is the y-intercept. This equation is used to describe the relationship between the x and y coordinates of points on the line.

## 2. How do you find the slope of a line?

The slope of a line is found by dividing the change in y-coordinates by the change in x-coordinates between two points on the line. It can also be calculated using the equation m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

## 3. What is multivariable calculus?

Multivariable calculus is a branch of calculus that deals with functions of more than one variable. It involves the study of limits, derivatives, and integrals of functions with multiple independent variables, and is used to solve problems in fields such as physics, engineering, and economics.

## 4. How do you find the equation of a line in multivariable calculus?

In multivariable calculus, the equation of a line is represented in the form z = ax + by + c, where a, b, and c are constants and x and y are the independent variables. This equation can be found using the point-slope form or the two-point form, similar to finding the equation of a line in two dimensions.

## 5. What is the significance of equations of lines in multivariable calculus?

Equations of lines are used in multivariable calculus to represent linear relationships between multiple variables. They are also used to calculate derivatives and integrals of multivariable functions, which are important concepts in the study of functions with multiple independent variables.

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