Question 172787
Start with the given system of equations


{{{system(x-y+2z=-3, x+2y+3z=4, 2x+y+z=-3)}}} 



{{{x-y+2z=-3}}} Start with the first equation



{{{x=-3+y-2z}}} Add "y" to both sides. Subtract {{{2z}}} from both sides. Now let's call this equation 4.



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{{{2x+y+z=-3}}} Move onto the third equation



{{{y=-3-2x-z}}} Subtract 2x from both sides. Subtract "z" from both sides. 



{{{y=-3-2(-3+y-2z)-z}}} Plug in {{{x=-3+y-2z}}} (equation 4).



{{{y=-3+6-2y+4z-z}}} Distribute



{{{y=3-2y+3z}}} Combine like terms.



{{{y+2y=3+3z}}} Add 2y to both sides.



{{{3y=3+3z}}} Combine like terms.



{{{y=(3+3z)/3}}} Divide both sides by 3 to isolate "y"



{{{y=(3)/3+(3z)/3}}} Break up the fraction.



{{{y=1+z}}} Reduce



{{{y=z+1}}} Rearrange the terms. Let's call this equation 5.
 


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{{{x+2y+3z=4}}} Now move onto the second equation



{{{(-3+y-2z)+2(z+1)+3z=4}}} Plug in {{{x=-3+y-2z}}} (equation 4) and {{{y=z+1}}} (equation 5)



{{{-3+y-2z+2z+2+3z=4}}} Distribute



{{{-1+y+3z=4}}} Combine like terms.



{{{y+3z=4+1}}} Add 1 to both sides.



{{{y+3z=5}}} Add. Now let's call this equation 6. 



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Are you with me so far? Well after all that substitution and simplification, we have the equations


{{{x=-3+y-2z}}} Equation 4



{{{y=z+1}}} Equation 5



{{{y+3z=5}}} Equation 6




So let's solve the system of equations of 5 and 6  

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{{{y+3z=5}}} Start with equation 6



{{{z+1+3z=5}}} Plug in {{{y=z+1}}} (Equation 5).



{{{1+4z=5}}} Combine like terms on the left side.



{{{4z=5-1}}} Subtract {{{1}}} from both sides.



{{{4z=4}}} Combine like terms on the right side.



{{{z=(4)/(4)}}} Divide both sides by {{{4}}} to isolate {{{z}}}.



{{{z=1}}} Reduce. 


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{{{y=z+1}}} Go back to Equation 5.



{{{y=1+1}}} Plug in {{{z=1}}}



{{{y=2}}} Add.


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{{{x=-3+y-2z}}} Go back to equation 4.



{{{x=-3+2-2(1)}}} Plug in {{{y=2}}} and {{{z=1}}}



{{{x=-3+2-2}}} Multiply



{{{x=-3}}} Combine like terms.




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Answer:



So the solutions are {{{x=-3}}}, {{{y=2}}}, and {{{z=1}}}


These solutions form the ordered triple (-3,2,1)