Solving Systems of Linear Equations in Three Variables

Solving Systems of Linear Equations in Three Variables

A solution of a system of equations in three

variables in an ordered triple (x,y,z) that makes all three equation true.

Solving systems using elimination

x + y + z = 6 12x – y +3z = 9 2-x + 2y + 2z = 9 3

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

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

• Use a and 5 together to solve for z

• By using 4 and 6 together we can solve for y.

3y + 3z =15 4

3y +3(3) = 15 3y + 9 = 153y = 15- 9

3y = 6 Y = 2 7

• Substitute 6 and 7 into 1 to solve for x.

X + y + z = 6X + (2) + (3) =6

X + 5 = 6X = 1

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

Creamer’s Rule For A 3 x 3 System

• Let A be the coefficient matrix of this linear system:

Ax + by +cz =jDx + ey + fz = kGx + hy + iz = l

Example problem :D

5x + 5y + 7z = 215x + 7y + 9z = 237x + 9y + 11z = 25

Intimidating? Very…… but also very easy to solve! :D

first get rid of the variables, literally forget about them so that it looks like5 + 5 + 7 = 215 + 7 + 9 = 237 + 9 + 11 = 25

From here go to your calculator and press the “2nd” button and then hit “x^-1.” After you do this press the right arrow until you reach edit.

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 2nd button and the x^-1 button again (Remember hit 2nd 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]

We are almost done now……. WEWTNESS!!!! :Dfrom here simply hit 2nd 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 2nd 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 zx= -14.625y = -18.875z= 24.5