Solve the following system and interpret the result geometrically

[lamerali]

Solve the following system and interpret the result geometrically

3x + 4y + 5z - 18 = 0
2x - y + 8z - 13 = 0
-x + 17y + 25z + 11 = 0


My answer:
3x + 4y + 5z – 18 = 0 (1)
2x – y + 8z – 13 = 0 (2)
-x + 17y + 25z + 11 = 0 (3)

Multiply (1) by 2 and (2) by 3

6x + 8y + 10z – 36 = 0
6x – 3y + 24z – 39 = 0

Subtract the two equations

11y – 14z + 3 = 0 (4)

Multiply equation (3) by -3

3x – 51y – 75z – 33 = 0

Subtract the equation from (1)

55y + 80z + 15 = 0

Simplify

11y + 16z + 3 = 0 (5)

We now have the new system
11y – 14z + 3 = 0 (4)
11y + 16z + 3 = 0 (5)

Subtract (4) from (5)

30z = 0


this is as far as i got...i'm not sure if it is correct and if it is correct i don't know what it geometrically represents.
any guidance is appreciated
Thanks in advance

 

[Dick]

You're doing fine so far. Now finish by finding x and y. If you think of x,y,z as three dimensional coordinates then the graph of each of those equations is very simple geometrical figure. Can you name it? Solving them simultaneously is the same as finding the intersection in geometric language.

 

[lamerali]

Solve for z
z = 0

substitute z = 0 into equation (4)

11y – 14(0) + 3 = 0

11y + 3 = 0
y = 3/11

substitute y and z into equation (1) to find x

3x + 4(3/11) + 5(0) – 18 = 0
x = 5.6

Therefore the planes intersect at the point (5.6, 0.273, 0) and are parallel to the z axis.

am i getting somewhere?
THANKS

 

[Dick]

Solve for z
z = 0

substitute z = 0 into equation (4)

11y – 14(0) + 3 = 0

11y + 3 = 0
y = 3/11

substitute y and z into equation (1) to find x

3x + 4(3/11) + 5(0) – 18 = 0
x = 5.6

Therefore the planes intersect at the point (5.6, 0.273, 0) and are parallel to the z axis.

am i getting somewhere?
THANKS

You are getting somewhere, but you are being pretty sloppy on the way. Shouldn't y=(-3/11)? Once you have (x,-3/11,0) then, sure, you have three planes that intersect in that point. But why would say any of them is parallel to the z axis? What would the equation of a plane parallel to the z axis look like?

 

[lamerali]

i got confused for some reason i thought because the z value of the point was zero there was a plane parallel to the z-axis...
so in the end would i simply say:
substitute y and z into equation (1) to find x

3x + 4(-3/11) + 5(0) – 18 = 0
x = -6.4

Therefore the three planes intersect at the point (-6.4, 0.273, 0)

 

[Dick]

i got confused for some reason i thought because the z value of the point was zero there was a plane parallel to the z-axis...
so in the end would i simply say:
substitute y and z into equation (1) to find x

3x + 4(-3/11) + 5(0) – 18 = 0
x = -6.4

Therefore the three planes intersect at the point (-6.4, 0.273, 0)

Now you've got the x value wrong. :( Why don't you just express them all as fraction like -3/11. That way you don't have to worry about round off.

 

[lamerali]

3x + 4(-3/11) + 5(0) – 18 = 0
x = -70/11

Therefore the three planes intersect at the point (-70/11, -3/11, 0)

 

[Dick]

You are making the same mistake as with y. The SIGN is wrong. Figure out what you are doing wrong that is causing this to happen and don't do it again!

 

[lamerali]

ahhh...i wasn't switching the sign when I added it to the R.S so the point is (70/11, -3/11, 0)

 

[Dick]

Riggght.

 

[lamerali]

YAY! Thanks for the help!