please solve this system of equations?

Question

x + y - 2z = 5
2x + 3y + 4z = 2

I know there is 3 variables and only 2 equations. but that shouldn't stop you from getting a solution. It says find all solutions. so if the solution is dependent on a variable that's fine. i just need a statment of all solutions

Answer 1

Again, use matrices:

[1 1 -2 | 5]
[2 3 4 | 2]

Subtract twice the first row from the second:

[1 1 -2 | 5]
[0 1 8 | -8]

Subtract the second row from the first:

[1 0 -10 | 13]
[0 1 8 | -8]

So...

x = 10z + 13
y = -8z -8
z = free

Answer 2

3 variables and only 2 equations => we will need to keep one variable as a parameter. Say z. This means we want to express x and y in terms of z.

Express x from the first equation and plug it into the second one.

x = - y + 2z + 5

=> 2(- y + 2z + 5) + 3y + 4z = 2
-2y + 4z + 10 + 3y + 4z = 2
y = -8z - 8

Ok, we have y. Now plug that back into the expression for x:
x = - y + 2z + 5
x = - ( -8z - 8) + 2z + 5
x = 10z + 13

As we said, we keep z as a parameter. z=z. It can take any real value.

So all the solutions are:
(x, y, z) = (10z + 13, -8z - 8, z)

You can get particular solutions for fixed values of z. For example, if z = 0, the particular solution is (13, -8, 0).

Hope this helps.

Answer 3

Let's use z as a parameter and let's solve the system
by Cramer's rule.
We have
x + y = 5+ 2z
2x + 3y = 2-4z
The denominator determinant for Cramer's rule is
|1 1| =1,
|2 3|
so we have
x = | 5+2z 1|
......| 2-4z 3|
x = 13 + 10z.
Plugging this back into the first equation gives
y = -8-8z.
So all solutions are given by
(x,y,z) = (13+10z, -8-8z, z).

Answer 4

Solve one equation for a variable:
x = 2z - y + 5

Plug it into the other equation to find the solution:
2(2z - y + 5) + 3y + 4z = 2
4z - y + 5 + 3y + 4z = 2
2y + 8z = -3
y = -4z - 1.5

x = 2z - y + 5 = 2z - 4z - 1.5 + 5
x = -2z + 3.5

Since there are 3 unknowns and 2 equations, you get an infinite solution set. Any z value will give a corresponding x and y value:
x = -2z + 3.5
y = -4z - 1.5

Answer 5

Double the first equation, and add the two together, to get 4x + 5y = 12, so x = 3 - 5/4 * y

All points that are on that line will solve the system.