Question 825231
<pre>
Hi, there--

Problem:
Use Cramer's Rule to solve this system of linear equations: 
x + 2y = -7
2x = 4 + 5y

A Solution:

Cramer's Rule uses the determinant of the coefficient matrix related to this system of 
equations to find the solution.

First, set up the 2 x 2 coefficient matrix. Write both equations in Ax + By = C form.

x + 2y = -7
2x -5y = 4

Now set up the 2 x 2 matrix using the coefficients of the x and y terms.
{{{( matrix(2,2,1,2,2,-5))}}}

To find the determinant D, we use the formula,

{{{D= det( matrix(2,2,a,b,c,d))=ad-cb}}}

For your system we have
{{{D= det( matrix(2,2,1,2,2,-5))=(1)(-5)-(2)(2)=-5-4=-9}}}


Now we find two more determinants, {{{D[x]}}} and {{{D[y]}}}.

For {{{D[x]}}}, we substitute the values on the right side of our equations for the left column of the coefficient matrix.

{{{D[x]=det( matrix(2,2,-7,2,4,-5))=(-7)(-5)-(4)(2)=35-8=27}}}

For {{{D[y]}}}, we substitute the values on the right side of our equations for the right column of the coefficient matrix.

{{{D[y]=det( matrix(2,2,1,-7,2,4))=(1)(4)-(2)(-7)=4+14=18}}}

Cramer's Rule states that for the solution to our system of equations (x,y)

{{{x=D[x]/D=27/-9=-3}}} 
and
{{{y=D[y]/D=18/-9=-2}}}

So, the ordered pair (-3,-2) is the solution to your system of equations.

Let's check this point in our original equations to be sure.
x + 2y = -7
(-3) + 2(-2) = -7
-3 - 4  = -7
-7 = -7 
CHECK!

2x = 4 + 5y
2(-3) = 4 + 5(-2)
-6 = 4 - 10
-6 = -6 CHECK!

That's it. I hope this helps,
Mrs. Figgy

</pre>