Linear system with three variables?

Question

x + 2y -z = -1
x - 3y + z =2
2x + y +2z =6

Answer 1

There are many ways to solve this, I believe the easiest is via elimination. I will refer to the equations via numbering them 1,2, and 3.

(1) x + 2y -z = -1
(2) x - 3y + z =2
(3) 2x + y +2z =6

The first step is to choose two of the equations, our goal is to eliminate on of the variables, so lets choose the (1) and (2).

x + 2y -z = -1
x - 3y + z = 2

If we just add the two equations we see that the z cancels out:

(*) 2x - y = 1

put this off to the side for the moment we'll do more with this later.

Now we must eliminate the z in the 3rd equation, so choose either the 1st or 2nd original equation to do this. Lets choose the 1st one.

x + 2y -z = -1
2x + y +2z =6

In order to get rid of the z multiply the first equation by 2 then add it to the 3rd equation.

2x + 4y -2z = -2
2x + y +2z =6

(**) 4x +5y = 4

Now recall the equation we put aside, (*) 2x - y = 1

Now we have two equations and two unknowns, so you may from here solve for x and y via any method you wish. We'll stick with elimination.

So let's multiply 2x - y =1 by -2, then add.

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

7y = 2, which means y = 2/7
y=2/7

now use this in either of the two equations above.

4x +5(2/7) = 4
4x +10/7 =4
4x = 4-10/7 = 28/7 - 10/7 = 18/7
4x = 18/7
x = 9/14

Now use these two solutions in the first set of equations.
Choose the first equation: x + 2y -z = -1 and plug in our solutions.

(9/14) + 2(2/7) -z = -1
(9/14) + (4/7) -z = -1
(9/14) + (8/14) - z = -1
17/14 -z = -1
z = 31/14

Last step is to check your work, that is plug the solutions into all three equations and see if things work out as they should.

Answer 2

Add 1st two equations:-
2x - y = 1

2x + 4y - 2z = - 2
2x + y + 2z = 6--------ADD

4x + 5y = 4
2x - y = 1----------- X by 5

4x + 5y = 4
10x - 5y = 5-------ADD
14x = 9
x = 9 / 14

36 / 14 + 5y = 4
5y = 56/14 - 36/14
y = 4 / 14 = 2 / 7

9 / 14 - 6 / 7 + z = 2
z = 28/14 - 9/14 + 12/14
z = 31/14
x = 9/14
y = 2/7

Answer 3

Use the first 2 to eliminate z
2x - y = 1

Use hte first and third to eliminate z
4x + 5y = 4

Now eliminate x from these two equations
7y = 2

y = 2/7
x = 9/14
z = 31/14

Answer 4

x + 2y -z = -1--------->(1)
x - 3y + z =2---------->(2)
2x + y +2z =6----------->(3)
Adding equtions (1) & (2),
2x-y=1--------->(4)
Multiplying equations (1) by 3,
2x+4y-2z=-2----------(5)
Adding equations (3) & (5),
4x+5y=4------->(6)
Multiplying equations (4) by 5,
10x-5y=5--------->(7)
Adding equations (6) & (7),
14x=9
x=9/14
Sustituting the value of x in (7)
10(9/14)-5y=5
45/7-5y=5
-5y=5-45/7
-5y=-10/7
y=2/7
Substituting the values of x & y in (1)
9/14+4/7-z=-1
17/14-z=-1
-z=-1-17/14
-z=-31/14
z=31/14
x=9/14;y=2/7;z=31/14

Answer 5

y = 2/7
x = 9/14
z = 31/14

Answer 6

Have you learned matrices yet?
Anyway, isolate x in the first equation, plug it into the other two, so you have two equations with two variable. Sorry, I don't really want to work it out.

Answer 7

learn!