Question

How do you use the Cramer's Rule?

Answer 1

very carefully...


you use it to solve systems of linear equations.

would you like some help with a problem?

Answer 2

yea that would be great\[4x-5y=39\]\[3x+8y=-6\]

Answer 3

first, make sure your equations are in standard form...

yours are.

next, solve for the denominator:\[D=\left|\begin{matrix}4 & -5 \\ 3 & 8\end{matrix}\right|\] Do this first because if equals 0 then the system does not have a point solution. It will either have no solutions or infinite solutions. I will discuss those later.

Do you know how to compute this?

Answer 4

yes

Answer 5

now let's compute some more determinants...
\[D_x=\left|\begin{matrix}39 &-5 \\ -6 & 8\end{matrix}\right|\]See what I did there?
\[D_y=\left|\begin{matrix}4 &39 \\ 3 & -6\end{matrix}\right|\]See what I did there?

It's perfectly okay if these determinants are 0.

Now to find the point where these lines intersect...

The x coordinate of the intersection is:\[\frac{ D_x }{ D }\]and the y coordinate of the intersections is: \[\frac{ D_y }{ D }\]Thus the point of intersection is:\[\left( \frac{ D_x }{ D } ,\frac{ D_y }{ D }\right)\]

Since we divide by D, it's important that in not be 0. If it is, however, it can still tell us something...

Answer 6

ok

Answer 7

I spoke too soon... it will only be helpful when dealing with 2 dimensional systems (lines). It's not applicable to 3 D (planes) of higher dimensions (hyperplanes) so I will omit it.

Needless to say, if D is zero, you will either have 0 or an infinite number of solutions. You will need to use other methods to determine which case you have.

Did you follow what I did for Dx and Dy?

Answer 8

yea i believe so

Answer 9

good, then you're good to go!

Answer 10

awesome thanks

Answer 11

you're welcome!