Solve Using an Inverse Matrix 4x+4=y , y=6x

Problem

{[4*x+4=y],[y=6*x])

Solution

1. Rearrange the equations into the standard form a*x+b*y=c to prepare for matrix representation.

4*x−y=−4

−6*x+y=0

2. Write the system as a matrix equation A*X=B where A is the coefficient matrix, X is the variable column, and B is the constant column.

[[4,−1],[−6,1]]*[[x],[y]]=[[−4],[0]]

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

det(A)=(4)*(1)−(−1)*(−6)

det(A)=4−6=−2

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

A(−1)=1/(−2)*[[1,1],[6,4]]

A(−1)=[[−0.5,−0.5],[−3,−2]]

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

[[x],[y]]=[[−0.5,−0.5],[−3,−2]]*[[−4],[0]]

6. Compute the values for x and y through matrix multiplication.

x=(−0.5)*(−4)+(−0.5)*(0)=2

y=(−3)*(−4)+(−2)*(0)=12

Final Answer

(x,y)=(2,12)

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