SOLUTION: I am supposed to solve by using inverse matrices. Can you please help me? This is all one problem. On my worksheet all of the information is inside one left parenthesis vs. two as

Algebra ->  Linear-equations -> SOLUTION: I am supposed to solve by using inverse matrices. Can you please help me? This is all one problem. On my worksheet all of the information is inside one left parenthesis vs. two as       Log On


   

Question 136248: I am supposed to solve by using inverse matrices. Can you please help me? This is all one problem. On my worksheet all of the information is inside one left parenthesis vs. two as I have it shown here.
Thank you!

{7x - 8y = -21
{x - y = -2
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1 is to create your coefficient matrix

%28matrix%282%2C2%2C7%2C-8%2C1%2C-1%29%29

Now augment this matrix with an identity matrix of the same dimensions:
Identity Matrix: %28matrix%282%2C2%2C1%2C0%2C0%2C1%29%29
Augmented Matrix: %28matrix%282%2C4%2C7%2C-8%2C1%2C0%2C+1%2C-1%2C0%2C1%29%29

Now use Gauss-Jordan Row Reduction to transform the identity matrix to the left side of the augmented matrix. The two columns on the right of the result will be the Inverse Matrix

%28matrix%282%2C4%2C7%2C-8%2C1%2C0%2C+1%2C-1%2C0%2C1%29%29

Swap rows 1 and 2:
%28matrix%282%2C4%2C1%2C-1%2C0%2C1%2C7%2C-8%2C1%2C0%29%29

Multiply row 1 by -7, add the result to row 2, replace row 2 with that result:
row 1 times -7: matrix%281%2C4%2C-7%2C%2B7%2C0%2C-7%29
result plus row 2 matrix%281%2C4%2C0%2C-1%2C1%2C-7%29
new matrix: %28matrix%282%2C4%2C1%2C-1%2C0%2C1%2C0%2C-1%2C1%2C-7%29%29

Multiply row 2 by -1
new matrix: %28matrix%282%2C4%2C1%2C-1%2C0%2C1%2C0%2C1%2C-1%2C7%29%29

Add row 2 to row 1, replace row 1 with the result:
row 2 plus row 1: matrix%281%2C4%2C1%2C0%2C-1%2C8%29
new matrix: %28matrix%282%2C4%2C1%2C0%2C-1%2C8%2C0%2C1%2C-1%2C7%29%29

So the inverse matrix is %28matrix%282%2C2%2C-1%2C8%2C-1%2C7%29%29

Create the constants matrix from the constants in the original equations:
%28matrix%282%2C1%2C-21%2C-2%29%29

Multiply the Inverse matrix times the constants matrix:


And now we can say: %28matrix%282%2C1%2Cx%2Cy%29%29=%28matrix%282%2C1%2C5%2C7%29%29

Check the answer:
7%285%29-8%287%29=35-56=-21 True
5-7=-2 True
Both true, answer checks.

Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
Solution by Edwin:
Here is an easier way to find the inverse of a 2x2 matrix:

----------------------------------------------
I am supposed to solve by using inverse matrices. Can you please help me? This is all one problem. On my worksheet all of the information is inside one left parenthesis vs. two as I have it shown here.
Thank you!
7x - 8y = -21
x - y = -2

Write the system as:

7x+-+8y+=+-21
1x+-+1y+=++-2

Then write that as:



Now we must find the inverse of the 2x2 matrix %28+matrix%282%2C2%2C7%2C-8%2C1%2C-1%29+%29

To find the matrix of a 2x2 determinant:

%28+matrix%282%2C2%2C7%2C-8%2C1%2C-1%29+%29

1.  Find the value of its determinant:

abs%28+matrix%282%2C2%2C7%2C-8%2C1%2C-1%29%29+=+%287%29%28-1%29-%28-8%29%281%29+=+-7%2B8+=+1

2.  Swap the upper left and lower right elements of %28+matrix%282%2C2%2C7%2C-8%2C1%2C-1%29+%29:

%28+matrix%282%2C2%2C-1%2C-8%2C1%2C7%29+%29

3. Change the sign of the upper right and lower left elements.

%28+matrix%282%2C2%2C-1%2C8%2C-1%2C7%29+%29

4. Divide every element by the value of the determinant found
   in step 1.

%28+matrix%282%2C2%2C%28-1%29%2F1%2C8%2F1%2C%28-1%29%2F1%2C7%2F1%29+%29

%28+matrix%282%2C2%2C-1%2C8%2C-1%2C7%29+%29

-----------------

Next we left-multiply both sides of the equation



by the inverse %28+matrix%282%2C2%2C-1%2C8%2C-1%2C7%29+%29

and we get:

 

Next we multiply the first two matrices on the left:

 

Simplifying,





Now multiply the two matrices on the left:



Simplifying:



Now multiply the two matrices on the right:



Simplifying

+%0D%0A%28+matrix%282%2C1%2Cx%2Cy%29+%29+=+%28++matrix%282%2C1%2C21-16%2C21-14+%29%29+++

+%0D%0A%28+matrix%282%2C1%2Cx%2Cy%29+%29+=+%28++matrix%282%2C1%2C5%2C7+%29%29+++

So the solution is;  x=5,y=7

Edwin