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System of equations
Solving system of equations using matrices

Solving system of equations using the augmented matrix
Systems with two equations and two avariables can also be solved using matrices and the augmented matrix.

Steps to solve systems of equations using the augmented matrix
1. Form the augmented matrix of the system.
2. Use elementary row operations to transform the augmented matrix to row echelon form.
3. Write the system of equations that correspond to the echelon form matrix.
4. Use back substitution to solve the system.
5. Check the solution in the original equations.

That is, first, arrange the system in the following form:
$\left{ax+by=c\\a'x+b'y=c$

Next create a 2x3 matrix, placing the x coefficients in the 1st column, the y coefficients in the 2nd column and the constants in the 3rd column, with a line separating the 2nd and the 3rd column:

(This is the augmented matrix)

Finally, row reduce the 2×3 matrix using the elementary row operations. The result should be the identity matrix on the left side of the line and a column of constants on the right side of the line.
These constants are the solution to the system of equations.

Solve: $\left{x+y=1\\2x-3y=-1$
1. The augmented matrix is:
2. Transform the augmented matrix to row eachelon form:

3. The corresponding system is: $\left{x+y=2\\y=1$
4. Use back substitution to find x: x+1=2, then x=1. The solution set is (1,1)

Solve using the augmented matrix:

 $\left{-x+4y=47\\7x-5y=-30$ x= y=