Solve Using an Inverse Matrix 3y-5=x , x+y=115

Problem

{[3*y−5=x],[x+y=115])

Solution

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

x−3*y=−5

x+y=115

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,−3],[1,1]]*[[x],[y]]=[[−5],[115]]

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

det(A)=(1)*(1)−(−3)*(1)

det(A)=1+3=4

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

A(−1)=1/4*[[1,3],[−1,1]]

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

X=A(−1)*B

[[x],[y]]=1/4*[[1,3],[−1,1]]*[[−5],[115]]

6. Perform the matrix multiplication.

[[x],[y]]=1/4*[[(1)*(−5)+(3)*(115)],[(−1)*(−5)+(1)*(115)]]

[[x],[y]]=1/4*[[−5+345],[5+115]]

[[x],[y]]=1/4*[[340],[120]]

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

[[x],[y]]=[[85],[30]]

Final Answer

{[3*y−5=x],[x+y=115])⇒(x,y)=(85,30)

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