cramer's rule for 3 x 3 linear systems

Cramer’s Rule for 3 x 3 systems : 

Cramer’s Rule for 3 x 3 systems

Slide 2: 

Just like the 2 x 2 system, you have to find D, Dx , Dy, and Dz. D is the regular determinant, Dx is the determinant with the x-column replaced with the answer column, Dy is the determinant with the y-column replaced with the answer column, and Dz is the z-column replaced with the answer column. Let’s work with the following system: 2x+3y + 2z = 10 3x – 4y + 5z = -25 4x + 5y + 3z = 19

Regular Determinant : 

Regular Determinant To find the regular determinant, first write your system in matrix format, minus the answer column. Next, write the first two columns again, directly to the right of your third column.

Slide 4: 

To find your determinant, draw lines. Next, multiply the numbers on each line together and write down each answer. You have 6 lines, so you better have 6 answers.

Slide 5: 

Add the first three answers, and subtract the last three. (2)(-4)(3) + (3)(5)(4) + (2)(3)(5) - (2)(-4)(4) - (2)(5)(5) - (3)(3)(3) (-24) + (60) + (30) – (-32) – (50) – (27) = 21

Slide 6: 

Now find Dx. Remember to replace BOTH x-columns with the answer column.

Slide 7: 

Add the first three answers, and subtract the last three. (10)(-4)(3) + (3)(5)(19) + (2)(-25)(5) - (2)(-4)(19) - (10)(5)(5) - (3)(-25)(3) (-120) + (285) + (-250) – (-152) – (250) – (-225) = 42

Slide 8: 

Now find Dy. Remember to replace BOTH y-columns with the answer column.

Slide 9: 

Add the first three answers, and subtract the last three. (2)(-25)(3) + (10)(5)(4) + (2)(3)(19) - (2)(-25)(4) - (2)(5)(19) - (10)(3)(3) (-150) + (200) + (114) – (-200) – (190) – (90) = 84

Slide 10: 

Now find Dz. Replace the z-column with the answer column; don’t replace a second column on accident.

Slide 11: 

Add the first three answers, and subtract the last three. (2)(-4)(19) + (3)(-25)(4) + (10)(3)(5) - (10)(-4)(4) - (2)(-25)(5) - (3)(3)(19) (-152) + (-300) + (150) – (-160) – (-250) – (171) = -63

Finding the Answers : 

All that’s left now is to find x, y, and z. Just like the 2 x 2 systems, we divide the determinants. Finding the Answers