SOLUTION: Could you please help me solve the system of equations? 3x+3y+z=5 x-3y+2z=-13 8x-2y+3z=-8

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Question 151298: Could you please help me solve the system of equations?
3x+3y+z=5
x-3y+2z=-13
8x-2y+3z=-8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
system%283x%2B3y%2Bz=5%2Cx-3y%2B2z=-13%2C8x-2y%2B3z=-8%29


Add equations 1 and 2. Let's call the result equation 4:

3x%2B3y%2Bz=5
+x-3y%2B2z=-13
-----------------
4x%2B0y%2B3z=-8


Equation 4: 4x%2B3z=-8


2%283x%2B3y%2Bz%29=2%285%29 Multiply both sides of equation 1 by 2


6x%2B6y%2B2z=10 Multiply. Let's call this equation 5

-------




3%288x-2y%2B3z%29=3%28-8%29 Multiply both sides of equation 3 by 3


24x-6y%2B9z=-24 Multiply. Let's call this equation 6



Now add equations 5 and 6. Let's call the result equation 7:
6x%2B6y%2B2z=10
+24x-6y%2B9z=-24
-------------------
30x%2B0y%2B11z=-14

Equation 7: 30x%2B11z=-14


===================================

So we now have the two equations equation 4 and equation 7:

equation 4: 4x%2B3z=-8
equation 7: 30x%2B11z=-14




Start with the given system of equations:

system%284x%2B3z=-8%2C30x%2B11z=-14%29



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for z.




So let's isolate z in the first equation

4x%2B3z=-8 Start with the first equation


3z=-8-4x Subtract 4x from both sides


3z=-4x-8 Rearrange the equation


z=%28-4x-8%29%2F%283%29 Divide both sides by 3


z=%28%28-4%29%2F%283%29%29x%2B%28-8%29%2F%283%29 Break up the fraction


z=%28-4%2F3%29x-8%2F3 Reduce



---------------------

Since z=%28-4%2F3%29x-8%2F3, we can now replace each z in the second equation with %28-4%2F3%29x-8%2F3 to solve for x



30x%2B11highlight%28%28%28-4%2F3%29x-8%2F3%29%29=-14 Plug in z=%28-4%2F3%29x-8%2F3 into the first equation. In other words, replace each z with %28-4%2F3%29x-8%2F3. Notice we've eliminated the z variables. So we now have a simple equation with one unknown.



30x%2B%2811%29%28-4%2F3%29x%2B%2811%29%28-8%2F3%29=-14 Distribute 11 to %28-4%2F3%29x-8%2F3


30x-%2844%2F3%29x-88%2F3=-14 Multiply


%283%29%2830x-%2844%2F3%29x-88%2F3%29=%283%29%28-14%29 Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



90x-44x-88=-42 Distribute and multiply the LCM to each side



46x-88=-42 Combine like terms on the left side


46x=-42%2B88Add 88 to both sides


46x=46 Combine like terms on the right side


x=%2846%29%2F%2846%29 Divide both sides by 46 to isolate x



x=1 Divide



So the first part of our answer is: x=1






Since we know that x=1 we can plug it into the equation z=%28-4%2F3%29x-8%2F3 (remember we previously solved for z in the first equation).



z=%28-4%2F3%29x-8%2F3 Start with the equation where z was previously isolated.


z=%28-4%2F3%29%281%29-8%2F3 Plug in x=1


z=-4%2F3-8%2F3 Multiply


z=-4 Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



So the third part of our answer is: z=-4




3x%2B3y%2Bz=5 Go back to the first equation


3%281%29%2B3y%2B%28-4%29=5 Plug in x=1 and z=-4


3%2B3y-4=5 Multiply


3y-1=5 Combine like terms.

3y=5%2B1 Add 1 to both sides.


3y=6 Combine like terms on the right side.


y=%286%29%2F%283%29 Divide both sides by 3 to isolate y.


y=2 Reduce.



So the second part of the answer is y=2


-----------------Summary------------------------------

So our answers are:

x=1, y=2, and z=-4


They form the point (1,2,-4)