Solve Using an Inverse Matrix 15x+11y=32 , 7y-9x=8

Problem

{[15*x+11*y=32],[−9*x+7*y=8])

Solution

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

15*x+11*y=32

−9*x+7*y=8

2. Write the system as a matrix equation A*X=B

A=[[15,11],[−9,7]],X=[[x],[y]],B=[[32],[8]]

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

det(A)=(15)*(7)−(11)*(−9)

det(A)=105+99=204

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

A(−1)=1/204*[[7,−11],[9,15]]

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

X=A(−1)*B=1/204*[[7,−11],[9,15]]*[[32],[8]]

6. Perform the matrix multiplication.

X=1/204*[[(7)*(32)+(−11)*(8)],[(9)*(32)+(15)*(8)]]

X=1/204*[[224−88],[288+120]]

X=1/204*[[136],[408]]

7. Simplify the resulting fractions to find the values of x and y

x=136/204=2/3

y=408/204=2

Final Answer

[[x],[y]]=[[2/3],[2]]

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