SOLUTION: Solve {{{system(x+2*y-z = 5, 2*x-4*y+z = 0, 3*x+2*y+2*z = 3)}}}

Algebra ->  Graphs -> SOLUTION: Solve {{{system(x+2*y-z = 5, 2*x-4*y+z = 0, 3*x+2*y+2*z = 3)}}}       Log On


   

Question 142991: Solve system%28x%2B2%2Ay-z+=+5%2C+2%2Ax-4%2Ay%2Bz+=+0%2C+3%2Ax%2B2%2Ay%2B2%2Az+=+3%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system:

x%2B2y-z=5 Equation 1

2x-4y%2Bz=0 Equation 2

3x%2B2y%2B2z=3 Equation 3



Add equations 1 and 2 label the result Equation 4

x%2B2y-z=5 Equation 1

+ 2x-4y%2Bz=0 Equation 2
-----------------------------

3x-2y=5 Equation 4



Add 2*equation 1 and equation 3 and label the result Equation 5


2%2A%28x%2B2y-z%29=2%2A5 2*Equation 1

+3x%2B2y%2B2z=3 Equation 3
-------------------------------

5x%2B6y=13 Equation 5




Now we have the new system of equations


3x-2y=5 Equation 4

5x%2B6y=13 Equation 5




Let's solve this system by use of elimination



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






In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).


So lets eliminate x. In order to do that, we need to have both x coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the x coefficients equal in magnitude but opposite in sign, we need to multiply both x coefficients by some number to get them to an common number. So if we wanted to get 3 and 5 to some equal number, we could try to get them to the LCM.



Since the LCM of 3 and 5 is 15, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -3 like this:




5%283x-2y%29=5%285%29 Multiply the top equation (both sides) by 5
-3%285x%2B6y%29=-3%2813%29 Multiply the bottom equation (both sides) by -3




Distribute and multiply

15x-10y=25
-15x-18y=-39


Now add the equations together. In order to add 2 equations, group like terms and combine them

%2815x-15x%29%2B%28-10y-18y%29=25-39

Combine like terms and simplify



cross%2815x-15x%29-28y=-14 Notice how the x terms cancel out




-28y=-14 Simplify




y=-14%2F-28 Divide both sides by -28 to isolate y




y=1%2F2 Reduce



Now plug this answer into the top equation 3x-2y=5 to solve for x

3x-2y=5 Start with the first equation



3x-2%281%2F2%29=5 Plug in y=1%2F2




3x-2%2F2=5 Multiply




3x-1=5 Reduce



3x=5%2B1Add 1 to both sides


3x=6 Combine like terms on the right side


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



x=2 Divide




So the first two answers are

x=2 and y=1%2F2





x%2B2y-z=5 Go back to Equation 1


2%2B2%281%2F2%29-z=5 Plug in x=2 and y=1%2F2


2%2B1-z=5 Multiply


-z%2B3=5 Combine like terms on the left side


-z=5-3Subtract 3 from both sides


-z=2 Combine like terms on the right side


z=%282%29%2F%28-1%29 Divide both sides by -1 to isolate z



z=-2 Divide




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Answer:


So the solution is


x=2, y=1%2F2, and z=-2