Systems of Linear Equations and Matrices: Study Notes

Systems of Linear Equations with Two Variables

Definition and Graphical Representation

A linear equation in two variables has the general form $ax + by = c$, where $a$, $b$, and $c$ are constants. The graph of such an equation is a straight line in the Cartesian plane.

• Slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: For $2x + 3y = 6$, rearrange to $y = -\frac{2}{3}x + 2$.

System of Linear Equations

When two or more linear equations are considered together, they form a system of equations. The solution to the system is the set of variable values that satisfy all equations simultaneously.

• Square system: Number of equations equals number of variables.

• Non-square system: Number of equations does not equal number of variables.

• Example:

  • $x - y = 1$

  • $2x + 3y = 7$

  This is a square system (2 equations, 2 variables).

Solutions of a System of Equations

Definition

The solution to a system of equations is the set of variable values $(x, y)$ that make all equations true.

• Example: Determine if $(-2, 1)$ and $(-4, -3)$ are solutions to:

  • $x - y = -1$

  • $2x - y = -5$

  Solution:

  • Substitute $(-2, 1)$: $-2 - 1 = -3 \neq -1$ (not a solution)

  • Substitute $(-4, -3)$: $-4 - (-3) = -1$, $2(-4) - (-3) = -8 + 3 = -5$ (is a solution)

Solving Systems by Graphing

Intersection of Lines

Graphically, the solution to a system of two linear equations corresponds to the intersection point(s) of their lines.

• Case I: Lines intersect – Unique solution at the intersection point. The system is consistent.

• Case II: Lines are parallel – No solution; the system is inconsistent. Slopes are equal, but intercepts differ ($m_1 = m_2$, $b_1 \neq b_2$).

• Case III: Lines coincide – Infinitely many solutions; the system is consistent. Both equations represent the same line ($m_1 = m_2$, $b_1 = b_2$).

Classification Table

Lines	Intersection	Parallel	Coincident
Number of Solutions	1	None	Infinitely many
Consistent/Inconsistent	Consistent	Inconsistent	Consistent
Independent/Dependent	Independent	Independent	Dependent

Matrix Representation of Systems

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. Each row corresponds to an equation, and each column to a variable or the constants.

• Example: For the system

  • $5 - x - y = 2$

  • $2x + 5y = 7$

  • $-3x - y + 8x = 4$

  The augmented matrix is: $\begin{bmatrix} -1 & -1 & 0 & 2 \\ 2 & 5 & 0 & 7 \\ -3 & -1 & 8 & 4 \end{bmatrix}$

Row Echelon Form

A matrix is in row echelon form if:

• Any rows consisting entirely of zeros are at the bottom.

• The first nonzero entry in each row (the leading entry) is to the right of the leading entry in the row above.

Elementary Row Operations

• Interchange two rows.

• Multiply a row by a nonzero constant.

• Add or subtract a multiple of one row to/from another row.

Solving Systems Using Matrices

1. Write the augmented matrix.

2. Use row operations to reduce to row echelon form.

3. Back-substitute to find the solution.

• Example: Solve $\begin{bmatrix} 2 & -2 & -4 \\ 3 & 1 & 6 \end{bmatrix}$ Row operations yield $y = 2$, $x = 0$; solution is $(0, 2)$.

Matrix Operations

Definition of a Matrix

A matrix is a rectangular array of numbers called entries or elements. The size of a matrix is given by the number of rows ($m$) and columns ($n$), denoted as $m \times n$.

• Example: $\begin{bmatrix} 6 & 8 & 9 & 7 & 4 & 5 \\ 1 & 2 & 3 & 4 & 5 & 6 \\ 7 & 8 & 9 & 1 & 2 & 3 \\ 4 & 5 & 6 & 7 & 8 & 9 \end{bmatrix}$ is a $4 \times 6$ matrix.

• Row matrix: Size $1 \times n$.

• Column matrix: Size $m \times 1$.

Matrix Notation

• The $(i, j)$-entry of a matrix $A$ is denoted $a_{ij}$.

• Example: For $A = \begin{bmatrix} 8 & 6 & 4 \\ 3 & 7 & 9 \end{bmatrix}$, $a_{11} = 8$, $a_{12} = 6$, $a_{13} = 4$.

Matrix Operations

• Scalar Multiple: Multiply every entry by a scalar $c$. For $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, $cA = \begin{bmatrix} c a_{11} & c a_{12} \\ c a_{21} & c a_{22} \end{bmatrix}$.

• Negative of a Matrix: $-A$ is obtained by multiplying every entry by $-1$. For $A = \begin{bmatrix} 1 & 2 \\ 4 & -2 \end{bmatrix}$, $-A = \begin{bmatrix} -1 & -2 \\ -4 & 2 \end{bmatrix}$.

• Matrix Subtraction: $A - B$ is defined as $A + (-B)$. For $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ and $B = \begin{bmatrix} b_{ij} \end{bmatrix}$, $A - B = \begin{bmatrix} a_{ij} - b_{ij} \end{bmatrix}$.

• Zero Matrix: All entries are zero. $A + 0 = A$, $A - A = 0$.

Summary Table: Matrix Operations

Operation	Definition	Example
Scalar Multiple	Multiply every entry by a scalar	$2 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$
Negative	Multiply every entry by $-1$	$-\begin{bmatrix} 1 & 2 \\ 4 & -2 \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ -4 & 2 \end{bmatrix}$
Subtraction	Subtract corresponding entries	$\begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 2 & 3 \end{bmatrix}$
Zero Matrix	All entries are zero	$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Key Concepts and Applications

• Systems of linear equations are foundational for solving problems in calculus, engineering, and applied sciences.

• Matrix methods provide efficient tools for handling large systems.

• Understanding consistency, independence, and matrix operations is essential for advanced topics such as differential equations and computational methods.

Additional info: These notes expand on the original handwritten and slide content by providing formal definitions, examples, and structured tables for clarity and completeness.