Systems of Linear Equations with Two Variables

Definition and Graphical Representation

A linear equation in two variables has the general form , where , , and are constants. The graph of such an equation is a straight line in the Cartesian plane.

• Slope-intercept form: , where is the slope and is the y-intercept.

• Example: For , rearrange to .

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:

  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 that make all equations true.

• Example: Determine if and are solutions to:

  Solution:

  • Substitute : (not a solution)

  • Substitute : , (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 (, ).

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

Classification Table

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

  The augmented matrix is:

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 Row operations yield , ; solution is .

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 () and columns (), denoted as .

• Example: is a matrix.

• Row matrix: Size .

• Column matrix: Size .

Matrix Notation

• The -entry of a matrix is denoted .

• Example: For , , , .

Matrix Operations

• Scalar Multiple: Multiply every entry by a scalar . For , .

• Negative of a Matrix: is obtained by multiplying every entry by . For , .

• Matrix Subtraction: is defined as . For and , .

• Zero Matrix: All entries are zero. , .

Summary Table: Matrix Operations

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.