Solving Three Variable Equations: Finding x, y, and z

[smart_worker]

Homework Statement

find (x,y,z)

Homework Equations

3x - 4y + 7z = 0,-------------->first equation

2x - y - 2z = 0,---------------->second equation

3x³ - y³ + z³ = 18.---->third equation

The Attempt at a Solution

on subtracting first two equations i get x - 3y + 9z = 0

using this if i solve the both first two equations i get 5y - 20z = 0.

if i add the first 2 equations i get x - y -z =0

with this equation if i solve the first equation i get y = 10z
but if i solve 2nd one i get y=0.

 

[Ray Vickson]

Homework Statement

find (x,y,z)

Homework Equations

3x - 4y + 7z = 0,-------------->first equation

2x - y - 2z = 0,---------------->second equation

3x³ - y³ + z³ = 18.---->third equation

The Attempt at a Solution

on subtracting first two equations i get x - 3y + 9z = 0

using this if i solve the both first two equations i get 5y - 20z = 0.

if i add the first 2 equations i get x - y -z =0

with this equation if i solve the first equation i get y = 10z
but if i solve 2nd one i get y=0.

The best way is to proceed systematically; it may take a bit longer, but it is helpful in avoiding errors. So, from eq (2) we get $y = 2x - 2z.$ Putting this into equation (1) we have
0 = 3x - 4(2x - 2z) + 7z = -5x +15 z
So, $x = 15z / 5 = 3z$, and putting this into the expression for y we have $y = 2(3z) - 2z = 4z$. Now put $x = 3z, y = 4z$ into equation (3).

Note: we started solving for y in terms of z and z from eq. (2). We could equally well have started by solving for x in terms of y and z from eq. (1), etc., but the expressions would have been a bit more complicated. When in doubt, just forge ahead and do it.

 

[Mark44]

smart_worker,
Merely subtracting one equation from another to get a third equation isn't much help if the new equation still has three variables in it. A better way to go would be to add a multiple of one equation to the other so as to eliminate a variable. For example, you could add (-4) times the second equation to the first to get a new equation in only x and z.

Ray is suggesting a different approach. Since he has gone into more detail, I'll leave you to follow his suggestion.