Solve Using an Inverse Matrix 4x+2=5y-3 , y=3x-1

Problem

{[4*x+2=5*y−3],[y=3*x−1])

Solution

1. Rearrange the equations into standard form a*x+b*y=c by moving all variable terms to the left and constants to the right.

4*x−5*y=−5

−3*x+y=−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.

[[4,−5],[−3,1]]*[[x],[y]]=[[−5],[−1]]

3. Calculate the determinant of matrix A denoted as det(A) using the formula a*d−b*c

det(A)=(4)*(1)−(−5)*(−3)

det(A)=4−15=−11

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

A(−1)=1/(−11)*[[1,5],[3,4]]

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

X=A(−1)*B

[[x],[y]]=−1/11*[[1,5],[3,4]]*[[−5],[−1]]

6. Perform the matrix multiplication.

[[x],[y]]=−1/11*[[(1)*(−5)+(5)*(−1)],[(3)*(−5)+(4)*(−1)]]

[[x],[y]]=−1/11*[[−10],[−19]]

7. Simplify the resulting values for x and y

x=10/11

y=19/11

Final Answer

[[x],[y]]=[[10/11],[19/11]]

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