Section 2.2: Using Matrices to Solve Systems of Linear Equations with Two Unknowns

Part A. Setting Up a System & Doing Row Operations

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Part B. Solving by Pivoting

Part C. Row-Reduced Echelon Form

Part A: Setting Up a System & Doing Row Operations

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

Answer
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 xy = 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 [abc]. The augmented matrix of a whole system is then a matrix with one row for each equation in the system.

Example: Matrix Form of a System:

System of Equations	Matrix Form
x2y = 5 3x = 9	1 2 5 3 0 9

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 R12 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 3 R12 R2 3 0 9 3 6 3 3 0 9 Press the little pearl to see how we got that

What is the effect of performing the following operation?

	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

Part B. Solving by Pivoting

Part C. Row-Reduced Echelon Form