Question 77602: I am stuck on this problem! What do you do when 2 variables cancel out?
(elimination mehtod)
x + y -z=-2
2x - y + z=-5
-x + 2y - 3z=-4
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! x + y - z = -2
2x - y + z = -5
-x + 2y - 3z = -4
.
If you use the elimination method on the top two equations, by adding these two equations
the y terms and the z terms cancel out and what's left after the addition is:
.
3x = -7
.
Divide both sides by 3 and the result is x = -7/3
.
At this point you've already solved for x. Therefore, in any of the three equations
you can replace x by -7/3. But to solve for three variables you need to involve three
equations. So you need to make use of one of the top two equations along with the bottom
equation. You can substitute -7/3 for x in the top equation and you get:
.
(-7/3) + y - z = -2
.
Add 7/3 to both sides of this equation to get rid of the -7/3 on the left side and you get:
.
+y - z = 1/3
.
Now take the bottom of the three equations and substitute -7/3 for x to get:
.
-(-7/3) + 2y - 3z = -4
.
This simplifies to:
.
7/3 + 2y - 3z = -4
.
Subtract 7/3 from both sides to get rid of the 7/3 on the left side and the equation
becomes:
.
2y - 3z = -4 - 7/3
.
but -4 is equivalent to -12/3. With this change the right side of -12/3 -7/3 becomes -19/3.
Substitute this to convert the equation to:
.
2y - 3z = -19/3
.
So now you are ready to deal with the top equation (after -7/3 is substituted for x)
and the bottom equation (after -7/3 is also substituted for x). These two equations are:
.
+y - z = 1/3
2y - 3z = -19/3
.
You can double the top equation by multiplying all the terms (both sides) by 2 to get:
.
2y - 2z = 2/3 and
2y - 3z = -19/3
.
Subtract the two equations to eliminate the y variable and after the subtraction the resulting
equation is:
.
z = 21/3 = 7
.
At this point you know that two of the variables are x = -7/3 and z = 7. You can return
to any of the original three equations and substitute these values for x and z to solve
for y. For example, return to the top equation:
.
x + y - z = -2
.
Substitute for x and z to get:
.
-7/3 + y - 7 = -2
.
On the left side combine the -7/3 and the -7. Note that the -7 is equivalent to -21/3 so
when you combine it with -7/3 the results are:
.
-28/3 + y = -2
.
Add 28/3 to both sides to eliminate the -28/3 on the left side. This addition causes the
equation to become:
.
y = -2 + 28/3
.
But note that -2 is equivalent to -6/3. So the equation is:
.
y = -6/3 + 28/3
.
and the terms on the right side combine to give you:
.
y = 22/3
.
Now you have all three values ... x = -7/3, y = 22/3, and z = 7
.
Hope this helps you to understand that the fact that two of the variables can drop out
with the first elimination step, but that it can be handled by sticking with the general process.
It's unusual that this can happen, but it doesn't really change the rules of the game very
much.
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