Solve the following system:` {[2x+y+z=1],[-x-3y+z=9],[x-y-2z=7]}`

The are a number of ways to solve this using matrices. We can also use substitution or linear combinations.

(1) Substitution: Solve one of the equations for one of the variables. We will solve the first equation for z:

z=1-2x-y Now substitute this expression in place of z in the last two equations:

-x-3y+(1-2x-y)=9

x-y+2(1-2x-y)=7

-------------------------- Eliminate the parantheses and collect like terms:

-3x-4y=8

-3x-3y=5 It is easier to use combinations here: subtract the first equation from the second to get:

`y=-3`

Then `-3x+12=8==> -3x=-4 ==>x=4/3`

Now ` ``z=1-2x-y==>z=1-2(4/3)-(-3)==>z=1-8/3+3==>z=4/3`

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The solution is `x=4/3,y=-3,z=4/3`

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(2) Using linear combinations:

2x+y+z=1 I

-x-3y+z=9 II

x-y+2z=7 III

Take I-II ==> 3x+4y=-8

Take III-2I==> -3x-3y=5

Which is the same 2 equations in x and y we had above.