Finding the equation of the intersection

In summary, the given system of equations contains three planes with different coefficients. By manipulating the equations, it can be shown that equations 1 and 2 lead to a contradiction, indicating that there are no values of x, y, and z that satisfy all three planes. However, the book provides a solution using parametric equations, which suggests that there may be a solution. The solution involves choosing one of the equations and solving for one variable in terms of the others, and then substituting this value into one of the other equations. The resulting equations in two variables can then be solved for the remaining variables. Although equations 1 and 2 do not provide a valid solution, they are still used in the process of finding the parametric
  • #1
cdot9
1
0

Homework Statement


Plane 1:2x-3y+8z=7
Plane 2:x-8y+z=14
Plane 3:5x-14y+17z=28

Homework Equations


N/A


The Attempt at a Solution


I took plane 1 and subtracted (plane 2)x2 to get
13y+6z=-21 (I will refer to this as equation 1)

Then I took (plane 2)x5 and subtracted plane 3 to get
-26y-12z=32
which simplifies to
13y+6z=-16 (I will refer to this as equation 2)

Then I was stumped because the two equations contradict each other and state 0=-5
which leads me to believe that there is no values of x,y,z that can satisfy all planes

although the answers in the book say
"consistent [14/13,-21/13,0] + t[-61/13,-6/13,1]"

I understand how they get the answer I just don't understand why, if equation 1 and 2 do not provide a valid statement why do they continue and find the parametric equations

like let z = t
13y=-6t -21
y=-6/13t -21/13
then substitute y into plane 2 to get
x-8(-6/13z - 21/13) + z=14 and simplifies
x=61/13z +14/13
then z=0+t
which gets the correct parametric equations which I tested by using different values of t and the points do satisfy all planes
I guess the real question I have is why do they chose "equation 1" instead of "equation 2" which would yield a different result if you were to use that to fidn the parametric equations, which I'm assuming the parametric equations don't work and why do you continue after the elimination of equation 1 and 2 which results in 0=-5.
 
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  • #2
cdot9 said:

Homework Statement


Plane 1:2x-3y+8z=7
Plane 2:x-8y+z=14
Plane 3:5x-14y+17z=28

Homework Equations


N/A


The Attempt at a Solution


I took plane 1 and subtracted (plane 2)x2 to get
13y+6z=-21 (I will refer to this as equation 1)

Then I took (plane 2)x5 and subtracted plane 3 to get
-26y-12z=32
Hello cdot9. Welcome to PF !

5×14 - 28 = 42, not 32.
which simplifies to
13y+6z=-16 (I will refer to this as equation 2)

Then I was stumped because the two equations contradict each other and state 0=-5
which leads me to believe that there is no values of x,y,z that can satisfy all planes

although the answers in the book say
"consistent [14/13,-21/13,0] + t[-61/13,-6/13,1]"

I understand how they get the answer I just don't understand why, if equation 1 and 2 do not provide a valid statement why do they continue and find the parametric equations

like let z = t
13y=-6t -21
y=-6/13t -21/13
then substitute y into plane 2 to get
x-8(-6/13z - 21/13) + z=14 and simplifies
x=61/13z +14/13
then z=0+t
which gets the correct parametric equations which I tested by using different values of t and the points do satisfy all planes
I guess the real question I have is why do they chose "equation 1" instead of "equation 2" which would yield a different result if you were to use that to fidn the parametric equations, which I'm assuming the parametric equations don't work and why do you continue after the elimination of equation 1 and 2 which results in 0=-5.
 

1. What is the equation of intersection?

The equation of intersection represents the mathematical relationship between two or more lines, curves, or surfaces that intersect at a specific point or points. It can be written in various forms, such as slope-intercept form, point-slope form, or standard form.

2. How do you find the equation of intersection?

To find the equation of intersection, you must first identify the points of intersection between the given lines, curves, or surfaces. Then, you can use various methods such as substitution, elimination, or graphing to solve for the coordinates of the intersection point. Finally, you can plug in these coordinates into one of the forms mentioned in the previous answer to obtain the equation of intersection.

3. Can the equation of intersection be written in different forms?

Yes, the equation of intersection can be written in different forms depending on the given lines, curves, or surfaces. Each form has its own advantages and can be useful for different purposes. For example, the slope-intercept form is useful for graphing, while the point-slope form is useful for finding the equation of a line passing through a given point.

4. Are there any special cases when finding the equation of intersection?

Yes, there are a few special cases when finding the equation of intersection. These include parallel lines or identical lines, where there is no unique solution, and perpendicular lines, where the intersection point has a specific relationship with the given lines. In these cases, the equations of intersection may have a simpler or special form.

5. Can you use technology to find the equation of intersection?

Yes, you can use technology such as graphing calculators or computer software to find the equation of intersection. These tools can quickly graph the given lines, curves, or surfaces and calculate the coordinates of the intersection point. However, it is important to understand the mathematical concepts behind finding the equation of intersection before relying solely on technology.

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