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System of equations

Solving system of equations using matrices

The inverse matrix method
Systems with two equations and two avariables can also be solved using matrices and the inverse matrix.

First, arrange the system in the following form:
$\left{ax+by=c\\a'x+b'y=c$
Next, create two matrices from the given system of equations:

1. The coefficient matrix, created by using the coefficients of the variables involved. So for our system, the coefficient matrix is:
A=

2. The constant matrix. It is created from the constants on the right side of the equal signs. In our system, the constant matrix is:
B=
This is equivalent to writing:

which is equivalent to the original two equations (check the multiplication yourself).
Represented using capital letters, we can say that: A·X=B (called matrix equation).
Then solve for the matrix variable X by left-multiplying both sides of the above matrix equation (A·X=B) by A-1. That is:
A·X=B => A-1·A·X=A-1·B => X=A-1·B·.

Use matrices to solve the system: $\left{2x+3y=5\\3x+4y=7$

A= X= B=
To solve the system, we need the inverse of A. A-1=
So the solution of the system is given by X=A-1·B.
Then:

The solution is x=1, y=1

Solve using the inverse matrix:

 $\left{x-5y=10\\7x+5y=-10$ x= y=