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3.6 Solving Systems of Linear Equations in 3 Variables p. 177

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A system of lin. eqns. in 3 variables Looks something like: x+3y-z=-112x+y+z=15x-2y+3z=21 A solution is an ordered triple (x,y,z) that makes all 3 equations true.

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Steps for solving in 3 variables 1.Using the 1 st 2 equations, cancel one of the variables. 2.Using the last 2 equations, cancel the same variable from step 1. 3.Use the results of steps 1 & 2 to solve for the 2 remaining variables. 4.Plug the results from step 3 into one of the original 3 equations and solve for the 3 rd remaining variable. 5.Write the solution as an ordered triple (x,y,z).

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Solve the system. 1.x+3y-z=-11 2x+y+z=1 zs are easy to cancel! 3x+4y=-10 2. 2x+y+z=1 5x-2y+3z=21 Must cancel zs again! -6x-3y-3z=-35x-2y+3z=21 -x-5y=18 -x-5y=18 2(2)+(-4)+z=1 2(2)+(-4)+z=1 4-4+z=1 4-4+z=1 3. 3x+4y=-10 -x-5y=18 -x-5y=18 Solve for x & y. 3x+4y=-10-3x-15y+54-11y=44 y=- 4 y=- 43x+4(-4)=-10 x=2 x=2 (2, - 4, 1) x+3y-z=-112x+y+z=15x-2y+3z=21z=1

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2.2x+2y+z=5 4x+4y+2z=6 Cancel zs again. -4x-4y-2z=-104x+4y+2z=6 0=- 4 0=- 4 Doesnt make sense! No solution Solve the system. 1.-x+2y+z=3 2x+2y+z=5 zs are easy to cancel! -x+2y+z=3-2x-2y-z=-5-3x=-2x=2/3-x+2y+z=32x+2y+z=54x+4y+2z=6

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3. x+y=3 2x+2y=6 Cancel the xs. -2x-2y=-62x+2y=6 0=0 0=0 This is true. ¸ many solutions Solve the system. 1.-2x+4y+z=1 3x-3y-z=2 zs are easy to cancel! x+y=3 2.3x-3y-z=2 5x-y-z=8 Cancel zs again. -3x+3y+z=-25x-y-z=82x+2y=6-2x+4y+z=13x-3y-z=25x-y-z=8

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