Let us start by quickly reviewing some basic terms from the tutorial for Section 2.1.

Q Just what is a "system of linear equations in two unknowns?"

A First, a linear equation in two unknowns x and y is an equation of the form

ax + by = c

where a, b, and c are numbers, and where a and b are not both zero.

Examples: Linear Equations:

4x + 5y = 0	This has a = 4, b = 5, c = 0
x - y = 11	This has a = 1, b = -1, c = 11
4x = 3	This has a = 4, b = 0, c = 11

Second, a system of linear equations is just a collection of these beasts. To solve a system of linear equations means to find a solution (or solutions) (x, y) that simultaneously satisfies all of the equations in the system.

Example: System of Linear Equations:

4x + 5y = 40 x-y = 1

This is a system of two linear equations with solution x = 5, y = 4.

Setting Up a System of Linear Equations in Matrix Form

Simply put, the augmented matrix form of a single linear equation ax + by = c is just the single row matrix [a  b  c]. The augmented matrix of a whole system is then a matrix with one row for each equation in the system.

Examples: Matrix Form of a System:

System of Equations	Matrix Form
x - 2y = 5 3x         = 9	1 -2 5 3 0 9
-3x + 2y = 10 2x - y = 0
-x - y + z = 1 2x - y       = 5
	2 - 1 0 4 -2 0 1 5

Doing Row Operations

Here are three things you can do to a system of equations without effecting the solution:

1. Switch any two equations
2. Multiply both sides of any equation by a non-zero number
3. Replace any equation by its sum with another equation. More generally, you can replace an equation by, say, 4 times itself plus 5 times another equation.

Corresponding to these changes are the following row operations on an augemented matrix.

Row Operation	Example
1. Switch two rows We write R1R2 to indicate switching Row 1 and Row 2	1 -2 5 3 0 9 R1R2 3 0 9 1 -2 5
2. Multiply a row by a non-zero number For instance, write the instruction       3 R2 next to Row 2 to mean "Multiply row 2 by 3."	1 -2 5 3 0 9 3 R2 1 -2 5 9 0 27
3. Replace a row by a combination with another row For instance, write the instruction       3 R1-2 R2 next to Row 1 to mean: "Replace Row 1 by three times Row 1 minus twice Row 2. In words: "Three times the top minus twice the bottom."	1 -2 5 3R1-2R2 3 0 9 -3 -6 -3 3 0 9 Press here to see how we got that.

	-2	-2	0
	-1	1	3		R2 + 2 R1

	-5 -3 3 -1 1 3			-2 -2 0 3 5 3
	-2 -2 0 -5 -3 3

Now perform the indiated row operations and press "Check."

-1 0 1 1 2R1 + R2 2 3 0 -1 4 -1 1 1 R3 - 2R2

You can now go on to the next part of the tutorial for Section 2.2 by pressing "Next Tutorial" on the sidebar.