Solve Using an Inverse Matrix 2x-6y=7 , 5x+2y=10

Problem

{[2*x−6*y=7],[5*x+2*y=10])

Solution

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

A=[[2,−6],[5,2]]

X=[[x],[y]]

B=[[7],[10]]

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

det(A)=(2)*(2)−(−6)*(5)

det(A)=4+30=34

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

A(−1)=1/34*[[2,6],[−5,2]]

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

X=1/34*[[2,6],[−5,2]]*[[7],[10]]

5. Perform the matrix multiplication.

X=1/34*[[(2)*(7)+(6)*(10)],[(−5)*(7)+(2)*(10)]]

X=1/34*[[14+60],[−35+20]]

X=1/34*[[74],[−15]]

6. Simplify the resulting values to find x and y

x=74/34=37/17

y=−15/34

Final Answer

[[x],[y]]=[[37/17],[−15/34]]

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