Solve {3x-2y+2z=30, -x+3y-4z=-33, 2x-4y+3z=42}
Please just solve the set provided above!!!!
This will be a bit more involved than the systems with
two unknowns, but the process is the same. The plan of
attack is to use equations one and two to eliminate z.
That will leave an equation with x and y. Then, use
equations one and three to eliminate z again, leaving
another equation with x and y. Those two equations will
be used to eliminate x, leaving us with the value of y. I'll
number equations I intend to use later so you can refer back
to them. That's enough discussion for now.
1) 3x-2y+2z=30
2) -x+3y-4z=-33
3) 2x-4y+3z=42
Equation one; multiply by 2 so the z term has 4 as the coefficient.
3x - 2y + 2z = 30
2 * (3x - 2y + 2z) = 30 * 2
4) 6x - 4y + 4z = 60
Add equation two to equation four:
6x - 4y + 4z = 60
+(-x + 3y - 4z = -33)
----------------------
5x - y = 27
5) 5x - y = 27
Multiply equation one by 3. Watch the coefficient of z.
3 * (3x - 2y + 2z) = 30 * 3
6) 9x - 6y + 6z = 90
Multiply equation three by 2. Again, watch the coefficient of z.
2 * (2x - 4y + 3z) = 42 * 2
7) 4x - 8y + 6z = 84
Subtract equation seven from equation six.
9x - 6y + 6z = 90
-(4x - 8y + 6z = 84)
----------------------
5x + 2y = 6
8) 5x + 2y = 6
Subtract equation eight from equation five. Both equations
have 5 as the coefficient of x. We eliminate x this way.
5x - y = 27
-(5x + 2y = 6)
---------------
-3y = 21
-3y = 21
y = -7 <<<<<<<<<<<<<<<<<<<
At this point, I am confident that I followed the
correct procedures to arrive at the value for y. Use
that value to determine the value of x.
~~~~~~~~~~~~~~~
Plug y into equation five to find x.
5x - y = 27
5x - (-7) = 27
5x + 7 = 27
5x = 27 - 7
5x = 20
x = 4 <<<<<<<<<<<<<<<<<<<
Plug y into equation eight, too.
5x + 2y = 6
5x + 2(-7) = 6
5x - 14 = 6
5x = 6 + 14
5x = 20
x = 4 same value for x, confidence high
Proceed, solving for the value of z.
~~~~~~~~~~~~~~~
Plug both x and y into equation one. We will
solve for z.
Equation one:
3x - 2y + 2z = 30
3(4) - 2(-7) + 2z = 30
12 + 14 + 2z = 30
26 + 2z = 30
2z = 30 - 26
2x = 4
z = 2 <<<<<<<<<<<<<<<<<<<
Continue using the original equations to
check the values.
Equation two:
-x + 3y - 4z = -33
-(4) + 3(-7) - 4z = -33
-4 - 21 - 4z = -33
-25 - 4z = -33
-4z = -33 + 25
-4z = -8
z = 2 same value for z, looking good
Equation three:
2x - 4y + 3z = 42
2(4) - 4(-7) + 3z = 42
8 + 28 + 3z = 42
36 + 3z = 42
3z = 42 - 36
3z = 6
z = 2 satisfied with the results
We have performed several checks along the
way, thus proving all three of the values.
x = 4, y = -7 and z = 2