Solve Using an Inverse Matrix 3y=11-2x , 3x=y-11

Problem

{[3*y=11−2*x],[3*x=y−11])

Solution

1. Rearrange the equations into standard form a*x+b*y=c so that the variables align.

2*x+3*y=11

3*x−y=−11

2. Write the system as a matrix equation A*X=B where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

A=[[2,3],[3,−1]]

X=[[x],[y]]

B=[[11],[−11]]

3. Calculate the determinant of matrix A denoted as det(A)

det(A)=(2)*(−1)−(3)*(3)

det(A)=−2−9=−11

4. Find the inverse matrix A(−1) using the formula A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/(−11)*[[−1,−3],[−3,2]]

A(−1)=[[1/11,3/11],[3/11,−2/11]]

5. Solve for X by multiplying A(−1) and B

X=A(−1)*B

X=[[1/11,3/11],[3/11,−2/11]]*[[11],[−11]]

6. Perform the matrix multiplication to find the values of x and y

x=(1/11)*(11)+(3/11)*(−11)=1−3=−2

y=(3/11)*(11)+(−2/11)*(−11)=3+2=5

Final Answer

(x,y)=(−2,5)

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