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Simultaneous equations
Systems of 3 variable Equations

Systems of equations that have three variables are systems of planes. Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions.

To solve a system of, write the given system in matrix form. Write all the information contained in the system in the single augmented matrix and choose one of these method to solve the system:

• Gaussian Elimination: Take an augmented matrix for the system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained. In particular, we bring the augmented matrix to Row-Echelon Form. At that point, the solutions of the system are easily obtained.

• Matrix inverse method: Write the given system as a single matrix equation AX=B . Solve the equation obtained for the matrix variable X by left-multiplying both sides of the above matrix equation (AX=B) by the inverse of the matrix A. Then, and you have the solution.

• Cramer's Rule: To use this method, you have only to follow the steps:

1. Write the coefficient matrix of the system (call this matrix A); if it is square, you may continue, otherwise Cramer's rule is not applicable here.
2. Compute the determinant of the coefficient matrix, |A|; if |A| is not zero you may continue, otherwise Cramer's rule is not applicable here.
3. Suppose the first variable of the system is x. Then write the matrix Ax as follows: substitute the column of numbers to the right of the equal signs instead of the first (from the left) column of A. Now compute the determinant of Ax, that is |Ax|.
4. The value of x in the solution is now |Ax| / |A|.
5. Repeat steps 3, and 4 with the remaining variables. In each case substitute the column of numbers instead of the column of A that corresponds to the variable you are using. If the variables are x, y, and z, then the values will now be:

 x = |Ax| / |A| y = |Ay| / |A| z = |Az| / |A|

And the solution is

 (x, y, z) = ( |Ax| / |A| , |Ay| / |A| , |Az| / |A| )

Solve:

 $\left{5x+3y-5z=3\\-x-y+3z=1\\4x+y-z=4$ x = y = z =