Linear Algebra (1)

Any system of linear equations can be transformed into a row equivalent system of equations or in other words one with the same solution set by using elementary row operations.

Elementary Row Operations
1.) Interchange two rows.
2.) Scale a row by a non-zero constant.
3.) Add a multiple of row to another row.

Consider the following matrix

1  0  1
0  1  1

Here, x = 1 and y = 1. If the rows are switched then the solution remains the same. If I scale a row, say row two by a constant c, then cy = c which means that cy/c = c/c but c/c = 1. So, y = 1; the solution remains the same. Now if I add row 1 to row 2 and replace row 1 with the result, then

1  1  2
0 1  1

Row 1 is now x + y = 2. So, this line has a y-intercept of 2 and a slope of -1 but still intersects row 2 at (1,1) because 1 + 1 = 2; a solution to the equation. In fact, no matter the multiple c of row 2 when added to row 1, which results in x + cy = c + 1 will always ensure that (1,1) is a solution since (1) + c(1) = c + 1.