Question 1203437
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The button on wolframalpha.com for seeing the step-by-step solution does not work unless you have acquired special privileges on their site.  And the beginning of the step-by-step solution you can see shows a rigid solution method that might not be the easiest.<br>
And with a system of 3 equations I would not use substitution as the other tutor does... although that is a valid method.<br>
Given three linear equations in three variables, I would definitely use elimination instead of substitution.<br>
The general solution method for a system of any number of linear equations is to eliminate one variable at a time.<br>
Our equations are<br>
{{{3x+6y-z=7}}}  [1]
{{{2x-3y+5z=-12}}}  [2]
{{{8x+9y-2z=12}}}  [3]<br>
Use the "-z" in the first equation to eliminate the variable z in the other two equations, using elimination.<br>
{{{15x+30y-5z=35}}}  [1], multiplied by 5
{{{2x-3y+5z=-12}}}  [2]
{{{17x+27y=23}}}  [4] [the sum of those two equations]<br>
and<br>
{{{-6x-12y+2z=-14}}}  [1], multiplied by -2
{{{8x+9y-2z=12}}}  [2]
{{{2x-3y=-2}}}  [5] [the sum of those two equations]<br>
We have reduced the system of 3 equations in 3 variables to a system of 2 equations in 2 variables.  Use elimination again to eliminate one of the variables; solve for the remaining variable, and back substitute to find the final answer.<br>
{{{17x+27y=23}}}  [4]
{{{18x-27y=-18}}}  [5], multiplied by 9
{{{35x=5}}}  [the sum of those two equations]
{{{x=5/35 = 1/7}}}<br>
Substitute x = 1/7 in [5] and solve for y<br>
{{{2/7-3y=-2}}}
{{{2-21y=-14}}}
{{{-21y=-16}}}
{{{y=(-16)/(-21)=16/21}}}<br>
Substitute x = 1/7 and y = 16/21 in [1] and solve for z<br>
{{{3/7+6(16/21)-z=7}}}
{{{3/7+32/7-z=7}}}
{{{35/7-z=7}}}
{{{5-z=7}}}
{{{z=2}}}<br>
ANSWER: (x,y,z) = (1/7,16/21,-2)<br>