Question

Solve the system of equations by determinants for z only:
2x+y=3z=-2
x-4y+z=24
-3x-y+4z=0

Answer 1

I believe they call this cramer's rule, does that sound familiar?

Answer 2

yea i think

Answer 3

So cramer's rule is stated pretty simply as this:\[\LARGE x_i=\frac{\det( A_i)}{\det(A)}\]

What does it mean? Well in terms of this problem i just represents the index of your variable. So in this case we have: \[\LARGE x_1=x , \ x_2 = y, \ x_3=z\] Just because it's harder to count letters, but we can count numbers. So since we want z, we are really looking for x_3 in the formula, so let's replace all the i's with 3 and go ahead and put in z there: \[\LARGE z= \frac{\det(A_3)}{\det(A)}\]

Now this should all be recognizable except maybe one thing, what's that on top of the fraction? det(A_3) is almost the same as the normal det(A) except that we are replacing the 3rd column with the vector that the equation is equal to. And that's it!

Answer 4

I did not understand anything, just now. I'm sorry

Answer 5

@Kainui

Answer 6

Then read it, think about it, and ask me questions about what you don't understand taking one step at a time.

Answer 7

alright for starters, I'm not understanding the formula
\[z=\frac{ \det(A _{3}) }{ \det(A) }\]
@Kainui

Answer 8

I thought I had to use elimination/substitution.

Answer 9

Well if you had to use elimination/substitution then why would you say:

"Solve by determinants"

Answer 10

That's what I thought, until you mentioned Crammer's law

Answer 11

did not mean to post that again btw