Gaussian Elimination Two Variables

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In this section, we will learn how to use Gaussian Elimination to solve a system of linear equations in two variables. We will only look at systems with two equations and with two variables (x and y). Up to this point, we have only covered three methods to solve a linear system: graphing, substitution, and elimination. Now we will proceed into unchartered waters and begin to learn about matrices. A matrix is nothing more than an ordered array of numbers. If we wanted to use a matrix to solve a linear system, we would write each equation in standard form and then list the numerical information (coefficients and constants only) inside of a matrix. The matrix will have brackets on the outside. The matrix we will construct will have a vertical line to separate the coefficients from the constants. This type of matrix is known as the augmented matrix. We can manipulate the matrix using row operations. The following are row operations: 1) We can interchange any two rows, just like we can switch the order of which equation is on top and which equation is on the bottom. 2) We can multiply any row by a non-zero number, just like we can multiply any equation by a non-zero number and not change the solution. 3) We can multiply a row by a real number and add this to the corresponding elements of any other row. We will use these row operations to produce a matrix that is in row echelon form. This gives us 1’s down the diagonal and 0’s below. In this form, we are given one unknown and can use substitution to find the other. Additionally, we can work further in the matrix and place it into reduced row echelon form. This form gives us all solutions without any further work. Reduced row echelon form gives us 1’s down the diagonal and 0’s above and below.
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