Solve Using an Inverse Matrix y=3.2+x , 7.1-x=7.2y

Problem

{[y=3.2+x],[7.1−x=7.2*y])

Solution

1. Rearrange the equations into the standard form a*x+b*y=c to prepare for matrix representation.

−x+y=3.2

−x−7.2*y=−7.1

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.

[[−1,1],[−1,−7.2]]*[[x],[y]]=[[3.2],[−7.1]]

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

det(A)=(−1)*(−7.2)−(1)*(−1)

det(A)=7.2+1=8.2

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

A(−1)=1/8.2*[[−7.2,−1],[1,−1]]

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

[[x],[y]]=1/8.2*[[−7.2,−1],[1,−1]]*[[3.2],[−7.1]]

6. Perform the matrix multiplication.

x=1/8.2*((−7.2)*(3.2)+(−1)*(−7.1))=(−23.04+7.1)/8.2=(−15.94)/8.2≈−1.9439

y=1/8.2*((1)*(3.2)+(−1)*(−7.1))=(3.2+7.1)/8.2=10.3/8.2≈1.2561

7. Simplify the fractions to find the exact values.

x=−1594/820=−797/410

y=103/82

Final Answer

(x,y)=(−797/410,103/82)

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