Solve for x and y in the following matrix equation using elimination method
(1/2 -1/4) (x) =(2)
(2/5 1/6).( y). 6
To solve for x and y using the elimination method, we will multiply both sides of the matrix equation by the inverse of the coefficient matrix:
(1/2 -1/4) (x) = (2)
(2/5 1/6) (y) (6)
The inverse of the coefficient matrix is:
( 4/1 2/1 )
(-5/2 -5/1 )
Multiplying both sides of the equation by the inverse matrix, we get:
( 4/1 2/1 ) (1/2 -1/4) (x) = ( 4/1 2/1 ) (2)
(-5/2 -5/1 ) (2/5 1/6) (y) (-5/2 -5/1 ) (6)
Simplifying, we have:
( 1 0 ) (x) = ( 8 )
( 0 1 ) (y) ( -12 )
This gives us the following equations:
x = 8
y = -12
Therefore, the solution to the matrix equation is x = 8 and y = -12.
To solve the matrix equation using the elimination method, we will perform row operations to eliminate one variable at a time. Here are the steps to solve for x and y:
Step 1: Write the augmented matrix:
[1/2 -1/4 | 2]
[2/5 1/6 | 6]
Step 2: Multiply the first row by 2 to get rid of the fractions:
[1 -1/2 | 4]
[2/5 1/6 | 6]
Step 3: Multiply the second row by 5 to get rid of the fractions:
[1 -1/2 | 4]
[2 5/6 | 30]
Step 4: Multiply the first row by 6 and the second row by 2 to eliminate the x variable in the second row:
[6 -3 | 24]
[12 5/3 | 36]
Step 5: Subtract the first row from the second row:
[6 -3 | 24]
[6 8/3 | 12]
Step 6: Subtract the second row from the first row:
[0 -11/3 | 12]
[6 8/3 | 12]
Step 7: Divide the second row by 6 to isolate the variable y:
[0 -11/3 | 12]
[1 4/9 | 2]
Step 8: Multiply the second row by 11 to eliminate the y variable in the first row:
[0 -11/3 | 12]
[11 4/3 | 22]
Step 9: Add the second row to the first row:
[11 -7/3 | 34]
[11 4/3 | 22]
Step 10: Divide the first row by 11 to isolate the variable x:
[1 -7/33 | 34/11]
[11 4/3 | 22]
So the solution to the matrix equation is x = 34/11 and y = 22/33.