Question 253952
{{{3x+y=1}}} Start with the second equation.



{{{y=-3x+1}}} Subtract 3x from both sides.



{{{x+y-2z=0}}} Move back to the first equation



{{{x-3x+1-2z=0}}} Plug in {{{y=-3x+1}}}



{{{-2x+1-2z=0}}} Combine like terms.



{{{-2x-2z=-1}}} Subtract 1 from both sides.



{{{2x+2z=1}}} Multiply every term by -1



So let's make {{{2x+2z=1}}} equation 4 



{{{5x+3y+7z=2}}} Move onto the third equation



{{{5x+3(-3x+1)+7z=2}}} Plug in {{{y=-3x+1}}}



{{{5x-9x+3+7z=2}}} Distribute



{{{-4x-3+7z=2}}} Combine like terms.



{{{-4x+7z=2-3}}} Subtract 3 from both sides.



{{{-4x+7z=-1}}} Combine like terms. We'll make this equation 5.


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So we have the new system of equations:


{{{system(2x+2z=1,-4x+7z=-1)}}}



{{{2(2x+2z)=2(1)}}} Multiply the both sides of the first equation by 2.



{{{4x+4z=2}}} Distribute and multiply.



So we now have


{{{system(4x+4z=2,-4x+7z=-1)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(4x+4z)+(-4x+7z)=(2)+(-1)}}}



{{{(4x+-4x)+(4z+7z)=2+-1}}} Group like terms.



{{{0x+11z=1}}} Combine like terms.



{{{11z=1}}} Simplify.



{{{z=(1)/(11)}}} Divide both sides by {{{11}}} to isolate {{{z}}}.



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{{{4x+4z=2}}} Now go back to the first equation.



{{{4x+4(1/11)=2}}} Plug in {{{z=1/11}}}.



{{{4x+4/11=2}}} Multiply.



{{{11(4x+4/cross(11))=11(2)}}} Multiply both sides by the LCD {{{11}}} to clear any fractions.



{{{44x+4=22}}} Distribute and multiply.



{{{44x=22-4}}} Subtract {{{4}}} from both sides.



{{{44x=18}}} Combine like terms on the right side.



{{{x=(18)/(44)}}} Divide both sides by {{{44}}} to isolate {{{x}}}.



{{{x=9/22}}} Reduce.



{{{y=-3x+1}}} Go back to the previously isolated equation.



{{{y=-3(9/22)+1}}} Plug in {{{x=9/22}}}



{{{y=-5/22}}} Combine like terms.



So the solutions are {{{x=9/22}}}, {{{y=-5/22}}} and {{{z=1/11}}}.



Which form the ordered triple *[Tex \LARGE \left(\frac{9}{22},-\frac{5}{22}, \frac{1}{11}\right)].



This means that the system is consistent and independent.