Question

Solve the following system of equations.

2x – y + z = –3
2x + 2y + 3z = 2
3x – 3y – z = –4

Answer 1

(1)2x – y + z = –3
(2)2x + 2y + 3z = 2
(3)3x – 3y – z = –4




We could go a couple of different ways here, but lets do it i guess by the standard way. First lets choose to eliminate one of the variables. Lets take the first and third equation and eliminate z:

2x-y+z=-3 and 3x-3y-z=-4.

Now lets add them together:
2x-y+z=-3

+ 3x-3y-z=-4
---------------
5x-4y=-7

Now, we obviously cant stop here, since (1) we have one equation with two unknows, and also (2) we need to find a solution for all the equations, and we still havent dealt with that second equation.

Now what we will do is elimate the same variable as before, namely z, from any other pair of equations, therefore leaving us with a system of two equations and 2 unknows

Answer 2

Now lets take the the second and third equations. To elimiate z we will have to do something to that third equation, inorder to put ourselvesin a situation to eliminate z. What we should do is multiply the third equation by 3.
(3x – 3y – z = –4)*(3)

Which yields: 9x-9y-3z=-12

Now we imply add these two equation together (the third and second)

9x-9y-3z=-12
+2x + 2y + 3z = 2
-----------------
11x-7y=-10

Now we have simply a system of equations in two variables. Lets label these new equations:
(a)5x-4y=-7
(b)11x-7y=-10

Let solve this system

Answer 3

(a)5x-4y=-7
(b)11x-7y=-10

What can we do here? Well, i say we begin by eliminating the x variable. To do we can multiply equation (a) by -11:

(5x-4y=-7)*(-11)=-55x+44y=77

Now lets multiply equation (b) by 5:
(11x-7y=-10)*(5)=55x-35y=-50

Let add these equations:
-55x+44y=77
+55x-35y=-50
--------------
9y=27, divide both sides by 9 and get: y=3

Answer 4

Now let go back to equation (a) 5x-4y=-7, and lets plug the value that we found for y:

5x-4(3)=-7
=5x-12=-7
=5x=5
divide both sides by 5 and get: x=1

Answer 5

Finally, we must take our two know variables, that is: x=1 and y=3, back to one of the original equations.

Lets take equation (1)2x – y + z = –3.

Lets plug in our variables:

2(1)-(3)+z=-3

=2-3+z=-3
=-1+z=-3
which means that z=-2.

Now we have found all three variables: x=1, y=3, and z=-2.

Answer 6

To check, you should plug the variable we found into each of the three orginal equations, and see if each equation is true, but i leave that up to you :)