Solve Using an Inverse Matrix x-3y=-6 , 5x+3y=42

Problem

{[x−3*y=−6],[5*x+3*y=42])

Solution

1. Write the system as a matrix equation of the form A*X=B

A=[[1,−3],[5,3]]

X=[[x],[y]]

B=[[−6],[42]]

[[1,−3],[5,3]]*[[x],[y]]=[[−6],[42]]

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

det(A)=(1)*(3)−(−3)*(5)

det(A)=3+15=18

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

A(−1)=1/18*[[3,3],[−5,1]]

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

X=1/18*[[3,3],[−5,1]]*[[−6],[42]]

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

X=1/18*[[(3)*(−6)+(3)*(42)],[(−5)*(−6)+(1)*(42)]]

X=1/18*[[−18+126],[30+42]]

X=1/18*[[108],[72]]

6. Simplify the resulting vector.

x=108/18=6

y=72/18=4

Final Answer

(x,y)=(6,4)

Want more problems? Check here!