Solve by using the matrix method: [tex]$4 x-9 y+11=0$[/tex] and [tex]$6 y-3 x=8$[/tex].
Answer (2)
Rewrite the equations in standard form: 4 x − 9 y = − 11 and − 3 x + 6 y = 8 .
Express the system in matrix form A X = B , where A = [ 4 − 3 − 9 6 ] and B = [ − 11 8 ] .
Calculate the inverse of A: A − 1 = [ − 2 − 1 − 3 − 3 4 ] .
Find the solution X = A − 1 B , resulting in x = − 2 and y = 3 1 .
The solution to the system of equations is x = − 2 , y = 3 1 .
Explanation
Understanding the Problem We are given the following system of equations:
4 x − 9 y + 11 = 0 6 y − 3 x = 8
We need to solve this system using the matrix method.
Rewriting the Equations First, rewrite the equations in the standard form a x + b y = c :
4 x − 9 y = − 11 − 3 x + 6 y = 8
Matrix Representation Represent the system of equations in matrix form as A X = B , where:
A = [ 4 − 3 − 9 6 ] , X = [ x y ] , and B = [ − 11 8 ]
Determinant Calculation Calculate the determinant of matrix A:
d e t ( A ) = ( 4 ) ( 6 ) − ( − 9 ) ( − 3 ) = 24 − 27 = − 3
Inverse Matrix Find the inverse of matrix A:
A − 1 = d e t ( A ) 1 [ 6 3 9 4 ] = − 3 1 [ 6 3 9 4 ] = [ − 2 − 1 − 3 − 3 4 ]
Solving for X Calculate the solution X = A − 1 B :
X = [ − 2 − 1 − 3 − 3 4 ] [ − 11 8 ]
Matrix Multiplication Perform the matrix multiplication to find the values of x and y:
x = ( − 2 ) ( − 11 ) + ( − 3 ) ( 8 ) = 22 − 24 = − 2 y = ( − 1 ) ( − 11 ) + ( − 3 4 ) ( 8 ) = 11 − 3 32 = 3 33 − 32 = 3 1
Final Solution Therefore, the solution is x = − 2 and y = 3 1 .
Verification Verify the solution by substituting the values of x and y back into the original equations:
4 ( − 2 ) − 9 ( 3 1 ) + 11 = − 8 − 3 + 11 = 0 (Correct) 6 ( 3 1 ) − 3 ( − 2 ) = 2 + 6 = 8 (Correct)
Examples
Matrix methods are used in various fields like engineering, physics, and computer graphics. For instance, in structural engineering, matrix methods help analyze the forces and stresses in complex structures like bridges and buildings. By representing the structure as a matrix, engineers can solve for the unknown forces and ensure the structure's stability and safety. Similarly, in computer graphics, matrices are used to perform transformations such as scaling, rotation, and translation of objects in 3D space.
We solved the system of equations using the matrix method by rewriting the equations in standard form and then representing them in matrix format. We calculated the inverse of the coefficient matrix and found the solution to be x = − 2 and y = 3 1 . Verification showed that the values satisfy both original equations.
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