You can put this solution on YOUR website! 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: . 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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Next divide all terms on both sides by -2 to solve for y. The answer is:
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Remember that . 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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Subtract 2x from both sides to eliminate the 2x on the left side and get:
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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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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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Substitute the right side of the equations we got above for y and z. When you do this
equation becomes:
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This reduces to:
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Double all the terms to get rid of the denominator of 2 and you get:
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Combine the x terms on the left side to get:
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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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and if you divide both sides by -7 you find that:
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Now that we know we can return to the given top equation, substitute -2 for x and
solve for y to get:
. =
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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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and dividing both sides by - 2 your get that .
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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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Do the multiplication on the left side:
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Add 4 to both sides to eliminate the -4 on the left side and you get:
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Finally multiply both sides by -1 so that you are finding +z. You end up with:
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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.
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