Find the Variables [[2,7],[5,3]]x=[[-3,8],[7,-9]]

Problem

[[2,7],[5,3]]*X=[[−3,8],[7,−9]]

Solution

1. Identify the matrix equation as A*X=B where A=[[2,7],[5,3]] and B=[[−3,8],[7,−9]]

2. Calculate the determinant of matrix A to ensure it is invertible.

det(A)=(2)*(3)−(7)*(5)

det(A)=6−35=−29

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

A(−1)=1/(−29)*[[3,−7],[−5,2]]

4. Solve for X by multiplying both sides of the equation by A(−1) on the left, such that X=A(−1)*B

X=−1/29*[[3,−7],[−5,2]]*[[−3,8],[7,−9]]

5. Multiply the matrices A(−1) and B

X=−1/29*[[(3)*(−3)+(−7)*(7),(3)*(8)+(−7)*(−9)],[(−5)*(−3)+(2)*(7),(−5)*(8)+(2)*(−9)]]

X=−1/29*[[−9−49,24+63],[15+14,−40−18]]

X=−1/29*[[−58,87],[29,−58]]

6. Simplify the resulting matrix by dividing each element by −29

X=[[2,−3],[−1,2]]

Final Answer

[[2,7],[5,3]]*X=[[−3,8],[7,−9]]⇒X=[[2,−3],[−1,2]]

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