Solving a 3x3 system?

[queen delta]

Show me an example

[harsh ferry]

Sorry! I realized I forgot to add it at first

[queen delta]

Okay

Do you know the determinate

Like 2 x 2 matrix

Is ac-bd

[harsh ferry]

Yes yes

[queen delta]

What’s this case

Correct myself

It’s “determinant”

Here

[proven jungle]

Wow.

Not bad.

There are a lot of ways to solve a 3x3 system.

The basic one involves Substitution method and Elimination method.

[queen delta]

a(ei-fb)+b(di-gf)+c(dh-ge)

[proven jungle]

Higher level one involved Matrices, just like what @queen delta has shown here.

[queen delta]

I’ve only solved 2x2

And used their Cramer rule

ac-bd

Easy

[harsh ferry]

And I can still apply that here?

[queen delta]

Yes

[queen delta]

Use this below

[queen delta]

Give me what’s a= a number to I=

[proven jungle]

You can actually add the first equation and the second equation to eliminate variable x.

Then multiply the first equation by 2 and subtract it with the third equation to eliminate variable x.

You'll end up with two new equations cosist of only two variables, which are y and z.

Solve for y and z and then plug in the values to solve for x.

[queen delta]

Restarter

How can you apply the determinant to this

[harsh ferry]

I should mention instead of -28 it is supposed to be -2

But I will try both methods!

Thank you so much for the tips

[queen delta]

Ok

[proven jungle]

You will have to form some matrices for this.

[queen delta]

Should it be 3 x 4

[proven jungle]

You're using Cramer's Rule right?

[queen delta]

Yes

[proven jungle]

Then we're not using 3×4 matrix.

This would be our main matrix, D:
$\begin{bmatrix}
1 & -2 & 3 \
-1 & 3 & 0 \
2 & -5 & 5
\end{bmatrix}$

And we're going to find it's determinant.

$\begin{vmatrix}
1 & -2 & 3 \
-1 & 3 & 0 \
2 & -5 & 5
\end{vmatrix}$

[frosty tree]

Restarter

[proven jungle]

Then after that, We need to find $D_x$.\
This can be done by replacing the x column with the solution column.\
Which is going to be:
$\begin{bmatrix}
9 & -2 & 3 \
-4 & 3 & 0 \
17 & -5 & 5
\end{bmatrix}$

[frosty tree]

Restarter

[proven jungle]

Then you find it's determinant:\
$\begin{vmatrix}
9 & -2 & 3 \
-4 & 3 & 0 \
17 & -5 & 5
\end{vmatrix}$

[frosty tree]

Restarter

[proven jungle]

To find the value of x in this case, we divide $\abs{D_x}$ with $\abs{D}$.

[frosty tree]

Restarter

Restarter

[proven jungle]

Repeat the same process to find y and z.

[ruby cave]

please, learn gaussian elimination.. cramer's method is a hassle to work with 😭

even the problem is telling you to use a calculator to get the determinant

[proven jungle]

Me too.

Either elimination or gaussian elimination or gauss-jordan elimination method.

I wouldn't use crammer's rule unless specified.

[mystic shore]

yk last semester during my linear Algebra course I used to use a trick to solve these problems more efficiently.
here is the link for the trick : https://youtu.be/HRiANm61i0s

but I would suggest you to ask your instructor if he would allow this method. But I hope this helps : D

[proven jungle]

Basketweave technique.

[mystic shore]

Yup, I don't know the name, but yeah. Schools and educational institutions should endorse such techniques to promote efficient problem solving.

[proven jungle]

Alright.

[queen delta]

Ik how to do 2x2 Cramer rule

D=(ac-bd)

[mystic shore]

well yeah its simple and ez.
but this method is for 3x3 matrix.

[queen delta]

Ok

[harsh ferry]

Also I should say I was able to find the solution! Idk exactly who’s method it was, but I just did the pattern and exchanged some of the numbers. Then I used the det to find x, y, and z. But desmos rlly did most of the work

Thank you again to all of you