Solve Using an Inverse Matrix x+y=6 , 2x-3y=2

Problem

{[x+y=6],[2*x−3*y=2])

Solution

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

A=[[1,1],[2,−3]]

X=[[x],[y]]

B=[[6],[2]]

2. Calculate the determinant of matrix A denoted as det(A) using the formula a*d−b*c

det(A)=(1)*(−3)−(1)*(2)

det(A)=−3−2=−5

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

A(−1)=1/(−5)*[[−3,−1],[−2,1]]

A(−1)=[[3/5,1/5],[2/5,−1/5]]

4. Solve for X by multiplying the inverse matrix A(−1) by the constant matrix B such that X=A(−1)*B

X=[[3/5,1/5],[2/5,−1/5]]*[[6],[2]]

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

x=(3/5)*(6)+(1/5)*(2)=18/5+2/5=20/5=4

y=(2/5)*(6)+(−1/5)*(2)=12/5−2/5=10/5=2

Final Answer

[[x],[y]]=[[4],[2]]

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