Solve Using an Inverse Matrix y=3x+1 , 3y-4x=13

Problem

{[y=3*x+1],[3*y−4*x=13])

Solution

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

−3*x+y=1

−4*x+3*y=13

2. Write the system as a matrix equation A*X=B

A=[[−3,1],[−4,3]]

X=[[x],[y]]

B=[[1],[13]]

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

det(A)=(−3)*(3)−(1)*(−4)

det(A)=−9+4=−5

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

A(−1)=1/(−5)*[[3,−1],[4,−3]]

A(−1)=[[−0.6,0.2],[−0.8,0.6]]

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

X=[[−0.6,0.2],[−0.8,0.6]]*[[1],[13]]

6. Perform the matrix multiplication to find the values of x and y

x=(−0.6)*(1)+(0.2)*(13)=−0.6+2.6=2

y=(−0.8)*(1)+(0.6)*(13)=−0.8+7.8=7

Final Answer

(x,y)=(2,7)

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