Question 72925
Given:
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x-2y=2
2x-z=-2
x-y-2z=4
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A clue to this one is that x is the only variable that appears in all three of these 
equations.  Therefore let's solve the top equation for y in terms of x.  Then let's 
solve the middle equation for z in terms of x.  When we do those two things we can substitute
for y and z in the bottom equation.  If we do that, then the bottom equation will have only 
x as a variable and we can, therefore, solve that equation for x and back solve to get
y and z.
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That's the plan. Let's see how it develops.
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The first equation is: {{{x - 2y = 2}}}.  Let's solve it for y in terms of x.
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First subtract x from both sides to get:
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{{{-2y = -x + 2}}}
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Next divide all terms on both sides by -2 to solve for y.  The answer is:
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{{{y = (-x/-2) +(2/-2) = (x/2) - 1}}}
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Remember that {{{y = (x/2) + 1}}}. We'll use it later.
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Next solve the middle equation for z in terms of x.  Start with:
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{{{2x-z=-2}}}
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Subtract 2x from both sides to eliminate the 2x on the left side and get:
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{{{-z = -2x - 2}}}
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Next multiply all terms on both sides by -1 to change -z to + z.  The result is:
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{{{z = 2x + 2}}}
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Remember this also.  We'll be using it shortly.
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Next go to the bottom one of the three given equations. It is:
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{{{x-y-2z=4}}}
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Substitute the right side of the equations we got above for y and z.  When you do this 
equation becomes:
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{{{x - (x/2 - 1) -2*(2x+2)= 4}}}
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This reduces to:
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{{{x - x/2 + 1 -4x - 4 = 4}}}
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Double all the terms to get rid of the denominator of 2 and you get:
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{{{2x - x + 2 - 8x - 8 = 8}}}
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Combine the x terms on the left side to get:
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{{{-7x +2 -8 = 8}}}
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Combine the numbers on the left side to get -6 and then add +6 to both sides to get rid
of the number -10 on the left side.  The result is:
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{{{-7x = 14}}}
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and if you divide both sides by -7 you find that:
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{{{x = -14/7 = -2}}}
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Now that we know {{{x=-2}}} we can return to the given top equation, substitute -2 for x and
solve for y to get:
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{{{x-2y=2}}} = {{{-2 - 2y = 2}}}
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Add +2 to both sides to get rid of the - 2 on the left side.  The equation then becomes:
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{{{-2y = 4}}}
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and dividing both sides by - 2 your get that {{{y = -2}}}.
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Now take the value of x and substitute it into the middle one of the given equations
to get:
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{{{2x-z=-2}}} = {{{2*(-2) - z = -2}}}
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Do the multiplication on the left side:
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{{{-4 - z = -2}}}
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Add 4 to both sides to eliminate the -4 on the left side and you get:
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{{{ - z = +2}}}
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Finally multiply both sides by -1 so that you are finding +z.  You end up with:
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{{{z = -2}}}
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Interesting.  All three variables (x, y, and z) equal -2.  
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Hope this method helps you to understand the substitution method of working with three linear 
equations that have a total of three variables.