Question

Explain, in complete sentences, which method you would use to solve the following system of equations and why you chose that method. Provide the solution to the system.

x - 3y + 4z = -9
2x + y + 3z = 6
x - 4y + 3z = -9

Answer 1

@phi

Answer 2

I would almost always use elimination, because it is methodical, and less confusing.

write down the equations without the variables
1 -3 4 -9
2 1 3 6
1 -4 3 -9

Answer 3

now you want to zero out the 2 just below the 1.
to do that, you have to add -2 to it. to get the -2, multiply the top row by -2
and add that to the 2nd row. that becomes your new 2nd row

you do not change the first row

Answer 4

First, what is -2 times the first row ?

Answer 5

-2 6 -8 18

Answer 6

now add -2 6 -8 18 to the 2nd row
-2 6 -8 18
2 1 3 6
-------------
0 7 -5 24
that is your new 2nd row. the problem is now:

1 -3 4 -9
0 7 -5 24
1 -4 3 -9

the next step is to make the 1 in the last row zero. we must add -1.
to get the -1 (using the first row), multiply the 1st row by -1
-1 * first row = -1 * 1 -3 4 -9 = -1 +3 -4 +9
add that to the 3rd row
-1 +3 -4 +9
1 -4 3 -9
--------------
0 -1 -1 0 <-- this is your new 3rd row

we now have

1 -3 4 -9
0 7 -5 24
0 -1 -1 0

Answer 7

The next step is to zero out the -1 in the bottom row

1 -3 4 -9
0 7 -5 24
0 -1 -1 0

multiply the 2nd row by 1/7 to get 0 1 -5/7 24/7
add this to the bottom row
0 1 -5/7 24/7
0 -1 -1 0
--------------
0 0 -12/7 24/7

we now have
1 -3 4 -9
0 7 -5 24
0 0 -12/7 24/7

if you remember that there are variables x , y ,z in there, the bottom row stands for

0x + 0y -12/7 z = 24/7
or simply -12z/7 = 24/7
multiply both sides by -7/12
\[ -\frac{7}{12}\cdot -\frac{12}{7} z = \frac{24}{7} \cdot -\frac{7}{12} \]
you get z= -2

Answer 8

the middle row is short for
7y -5z= 24
but you know z= -2, so this is
7y + -5* -2 = 24
7y +10= 24
7y = 24-10 = 14
y= 14/7 =2
y=2

now use the 1st row to solve for x, using y=2 and z= -2
1 -3 4 -9 is short for
x + -3*y + 4*z = -9
I assume you can find x.