Solve Using an Inverse Matrix 7x-5y=12 , 2x+3y=9

Problem

{[7*x−5*y=12],[2*x+3*y=9])

Solution

1. Write the system in matrix form A*X=B where A is the coefficient matrix, X is the variable column vector, and B is the constant column vector.

A=[[7,−5],[2,3]]

X=[[x],[y]]

B=[[12],[9]]

2. Calculate the determinant of matrix A denoted as det(A) or |A|

det(A)=(7)*(3)−(−5)*(2)

det(A)=21+10=31

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

A(−1)=1/31*[[3,5],[−2,7]]

4. Solve for X by multiplying the inverse matrix A(−1) by the constant vector B

X=A(−1)*B

X=1/31*[[3,5],[−2,7]]*[[12],[9]]

5. Perform the matrix multiplication.

X=1/31*[[(3)*(12)+(5)*(9)],[(−2)*(12)+(7)*(9)]]

X=1/31*[[36+45],[−24+63]]

X=1/31*[[81],[39]]

6. Simplify the resulting vector to find the values of x and y

x=81/31

y=39/31

Final Answer

[[x],[y]]=[[81/31],[39/31]]

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