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Solving Systems of Linear Equations in Three Variables

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A solution of a system of equations in three variables in an ordered triple (x,y,z) that makes all three equation true.

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Solving systems using elimination x + y + z = 6 1 2x – y +3z = 9 2 -x + 2y + 2z = 9 3

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Add 1 and 3 together. This will cancel out an x- term

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Next, multiply 1 by -2, then add 1 and 2 together to eliminate the x- term.

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Use a and 5 together to solve for z

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By using 4 and 6 together we can solve for y. 3y + 3z =15 4 3y +3(3) = 15 3y + 9 = 15 3y = 15- 9 3y = 6 Y = 2 7

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Substitute 6 and 7 into 1 to solve for x. X + y + z = 6 X + (2) + (3) =6 X + 5 = 6 X = 1

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( 1, 2, 3 ) Write the solution as an ordered triple.

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Creamers Rule For A 3 x 3 System

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Let A be the coefficient matrix of this linear system: Ax + by +cz =j Dx + ey + fz = k Gx + hy + iz = l

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Example problem :D 5x + 5y + 7z = 21 5x + 7y + 9z = 23 7x + 9y + 11z = 25

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Intimidating? Very…… but also very easy to solve! :D first get rid of the variables, literally forget about them so that it looks like 5 + 5 + 7 = 21 5 + 7 + 9 = 23 7 + 9 + 11 = 25 From here go to your calculator and press the 2 nd button and then hit x^-1. After you do this press the right arrow until you reach edit.

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For the sake of understanding this we are going to pick to edit the letter A From here make it a 3 by 3 matrix and put in the values it should end up looking like this [5, 5, 7] [5, 7, 9] [7, 9, 11] After this we are going to hit the 2 nd button and the x^-1 button again (Remember hit 2 nd first before hitting x^-1 otherwise you will mess everything up). From here go to EDIT and pick B This time make it a 3 by 1 matrix and make it look like [21] [23] [25]

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We are almost done now……. WEWTNESS!!!! :D from here simply hit 2 nd and go to x^-1 and instead of going to edit just press the number 1 and from here press x^-1 WITHOUT PRESSING THE 2 nd BUTTON LEAVE THAT ALONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! After you have done that press enter and you should get [.5, -1,.5] [-1, -.75, 1.25] [.5, 1.25, -1.25] from here multiply this by the matrix B to get [-14.625] [-18.875] [24.5] And from here you have your answers to the variables x, y and z x= -14.625 y = -18.875 z= 24.5

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6.3 – Solving Linear Systems By Linear Combination.

6.3 – Solving Linear Systems By Linear Combination.

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