Solve Using an Inverse Matrix y=6-2x , 7x-y=3

Problem

{[y=6−2*x],[7*x−y=3])

Solution

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

2*x+y=6

7*x−y=3

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.

A=[[2,1],[7,−1]]

X=[[x],[y]]

B=[[6],[3]]

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

det(A)=(2)*(−1)−(1)*(7)

det(A)=−2−7=−9

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

A(−1)=1/(−9)*[[−1,−1],[−7,2]]

A(−1)=[[1/9,1/9],[7/9,−2/9]]

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

X=[[1/9,1/9],[7/9,−2/9]]*[[6],[3]]

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

x=(1/9)*(6)+(1/9)*(3)=6/9+3/9=9/9=1

y=(7/9)*(6)+(−2/9)*(3)=42/9−6/9=36/9=4

Final Answer

(x,y)=(1,4)

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