This module covers the use of linear relations, including piecewise-defined relations, and non-linear relations to model a range of practical situations and solve related problems, including optimisation problems by linear programming.

Construction and interpretation of graphs, including:

- straight-line graphs, line segment graphs and step graphs and their use to model and analyse practical situations
- simultaneous linear equations in two unknowns and their use to model and analyse practical situations including break-even analysis, where cost and revenue functions are linear
- non-linear graphs and their use to model and analyse practical and familiar situations including the practical significance and interpretation of intercepts, slope, maximum/minimum points and the average rate of change when interpreting the graph
- non-linear graphs, either constructed from a table of data or given, the use of interpolation and extrapolation to predict values, estimation of maximum/minimum values and location; and coordinates of points of intersection for applications such as break-even analysis with non-linear cost and revenue functions
- graphical representation of relations of the form of \( y = kx^n \) for \( x \ge 0\), where \( n ∈ {–2, –1, 1, 2, 3} \), and their use in modelling practical situations including the determination of the constant of proportionality \( k \) by substitution of known values or by plotting \( y \) against \( x^n \) to linearise a given set of data, and the use of linearisation to test the validity of a proposed model.

Linear programming, including:

- review of linear inequalities in one and two variables and their graphical representation
- graphs of systems of linear inequalities (no more than five including those involving only one variable) and the use of shading-in to identify a feasible region
- linear programming and its purpose
- formulation of a linear programming problem including the identification of the decision variables, the construction of a system of linear inequalities to represent the constraints, and the expression of the quantity to be optimised (the objective function) in terms of the decision variables
- use of the graphical method to solve simple linear programming problems with two decision variables, and the sliding-line method and the corner-point principle as alternative methods for identifying optimal solutions
- extension of the linear programming method to include problems where integer solutions are required (for feasible regions containing only a small number of possible integer solutions only).

*source – VCE Mathematics Study Design*

Lessons

#### Gradient of Lines

- Video - Understanding Gradient Formula (3:56)
- Video - Finding Gradients of Straight Lines (2:56)
- Video - Applications of Gradients (6:38)
- Video - Gradient from Two Points (4:10)
- Video - Gradient from an Angle (4:03)
- Video - Finding the Positive Angle of x-axis (4:10)
- Video - Finding Gradients (4:55)
- Video - Gradient Undefined (1:21)
- Video - Gradient Zero (1:32)
- Video - Gradient from Rise and Run (4:48)

#### Intercepts

- Video - Straight Line Equations with Intercepts & Points (5:47)
- Video - Graphing Linear Equations using Intercepts (3:25)
- Video - Sketching Straight Lines using Intercepts (6:03)
- Topic - x-intercepts of Straight Lines in Standard Form
- Topic - y-intercepts of Straight Lines in Standard Form
- Topic - x-intercepts of Straight Lines in General Form
- Topic - y-intercepts of Straight Lines in General Form
- Topic - Gradients using x- and y-intercepts

#### Parallel Lines

- Video - Parallel Lines (4:16)
- Video - Identifying Parallel Lines (4:49)
- Video - Application of Parallel Lines (4:50)
- Video - Parallel Lines involving Similar Figures (9:46)
- Video - Equations of Parallel Lines Passing Through Points (4:27)
- Video - Equations of Parallel Lines in General Form Passing Through Points (4:25)
- Video - Equations of Parallel Lines in General form Passing through Midpoints (4:06)
- Video - Equations of Parallel Lines with Angles Passing Through Points (2:51)
- Video - Proof of Parallel Lines (2:58)
- Topic - Matching Parallel Lines

#### Perpendicular Lines

- Video - Finding the Gradients of Perpendicular Straight Lines (5:59)
- Video - Matching Perpendicular Lines (2:28)
- Video - Perpendicular Lines (4:44)
- Video - Perpendicular Lines in General Form (8:36)
- Video - Perpendicular Lines involving x-intercepts & y-intercepts (3:16)
- Video - Perpendicular Lines involving Angles (1:40)
- Video - Finding Equations of Perpendicular Lines (2:36)
- Topic - Gradient of a Straight Line which is Perpendicular to
- Topic - Choose the Straight Lines which is Perpendicular to
- Topic - Unknown Values Determining Two Straight Lines are Perpendicular
- Topic - Perpendicular Gradient from General Form
- Topic - Perpendicular Gradient involving x & y-intercepts
- Topic - Perpendicular Gradient involving Angles

#### Application of Perpendicular Lines

- Video - Applications involving Perpendicular Lines (6:36)
- Video - Finding Unknowns in Perpendicular Line Equations (2:55)
- Video - Equation of Lines Perpendicular to Line Joining Two Points (3:38)
- Video - Perpendicular Lines involving Triangles (8:47)
- Video - Perpendicular Lines involving Right-Angled Triangles (13:09)
- Video - Perpendicular Lines involving Locus (3:43)
- Video - Perpendicular Lines involving Angles Between Lines (9:10)

#### Vertical Lines and Horizontal Lines

- Video - Horizontal Lines and Vertical Lines (2:25)
- Video - Equations of Vertical Lines & Horizontal Lines (2:57)
- Topic - Graphs of Vertical Lines Horizontal Lines
- Topic - Equations of Vertical Lines Horizontal Lines
- Topic - Coordinates of Vertical Lines Horizontal Lines
- Topic - Gradients of Vertical Lines Horizontal Lines
- Topic - Intersections of Vertical Lines Horizontal Lines
- Topic - Status of Vertical Lines Horizontal Lines

#### Finding Equations of Lines

- Video - Introduction of Gradients & y-Intercepts (4:39)
- Video - Equation of Straight Lines involving Gradient & y-intercept (1:09)
- Video - Equations of Straight Lines using Gradients & Passing Points (7:57)
- Video - Point-Gradient Formula (3:09)
- Video - Linear Equations using Two Point Formula (2:55)
- Video - Two-Point Formula (6:17)
- Video - Equations of Straight Lines using Gradients & y-Intercepts (2:16)
- Video - Equation of Straight Lines involving Gradient & Origin (0:43)
- Video - Equation of Straight Lines involving Gradient & Points (1:57)
- Video - Intercept Form of Straight Lines (2:41)
- Video - Equations of Straight Lines using Gradients & Points (2:05)
- Video - Equations of Straight Lines using Gradients & Midpoints (1:30)

#### General Form of a Line

- Video - General Form & Gradient-Intercept Form of Straight Lines (6:35)
- Video - General Form of Straight Lines (3:28)
- Video - Equations of Straight Lines using Gradients & Points in General Form (3:05)
- Video - Gradient-Intercept Form (1:42)
- Video - Sketching Straight Lines using Intercepts in General Form (3:22)

#### Finding a Linear Equation from a Graph

- Video - Finding Equation of Lines (4:49)
- Video - Finding Linear Equations from Straight Line Graphs Passing Origin (3:22)
- Video - Finding Linear Equations from Straight Line Graphs (3:14)
- Video - Finding Linear Equations from Graphs (6:10)
- Video - Finding Linear Equations from Graphs Passing Origin (2:40)
- Video - Finding Linear Equations from Horizontal & Vertical Lines (1:31)
- Video - Equations of Straight Lines using Two Points (4:20)

#### Sketching Linear Graphs

- Video - Graphing Linear Equations using Gradients 1 (3:24)
- Video - Graphing Linear Equations using Gradients 2 (3:36)
- Video - Sketching Straight Lines using Gradients (4:26)
- Video - Straight Line Graphs in Positive Gradient Passing Origin (4:02)
- Video - Straight Line Graphs in Negative Gradient Passing Origin (3:02)
- Video - Straight Line Graphs in General Form Passing Origin (3:14)
- Video - Straight line Graphs in Positive Gradient by x- and y-intercepts (4:07)
- Video - Straight line Graphs in Negative Gradient by x- and y-intercepts (2:47)
- Video - Straight line Graphs in General Form by x- and y-intercepts (1:58)

#### Finding Intersection of Lines Graphically

- Video - Points of Intersections between Two Straight Lines (4:03)
- Video - Points of Intersections between Two Straight Lines in General Form (4:58)
- Video - Simultaneous Equations by Elimination Method & Graphs 1 (4:25)
- Video - Simultaneous Equations by Elimination Method & Graphs 2 (2:56)
- Video - Simultaneous Equations with Graphs (8:50)
- Video - Undefined in Simultaneous Equations with Graphs (2:49)
- Video - All Real Values in Simultaneous Equations with Graphs (2:50)

#### Linear Simultaneous Equations - Elimination Method

- Video - Elimination Method for Simultaneous Equations (3:32)
- Video - Elimination Method for Simultaneous Equations by Subtraction (4:29)
- Video - Elimination Method for Simultaneous Equations by Addition (3:25)
- Video - Elimination Method by Manipulating One Pronumeral (6:40)
- Video - Elimination Method by Manipulating Two Pronumerals 1 (3:31)
- Video - Elimination Method by Manipulating Two Pronumerals 2 (1:49)
- Video - Elimination Method by Manipulating Two Pronumerals 3 (2:15)
- Video - Simultaneous Equations by Elimination Method (7:42)
- Video - Simultaneous Equations: Elimination Method (5:53)
- Topic - Elimination Method by Subtracting Equations
- Topic - Elimination Method by Adding Equations
- Topic - Elimination Method by Considering One Coefficient
- Topic - Elimination Method by Considering Two Coefficients

#### Linear Simultaneous Equations - Substitution Method

- Video - Equation Method for Simultaneous Equations (2:16)
- Video - Basic Substitution Method for Simultaneous Equations (6:43)
- Video - Substitution Method for Simultaneous Equations (7:48)
- Video - Simultaneous Equations by Substitution Method (5:26)
- Video - Simultaneous Equations: Substitution Method (3:00)
- Topic - Basic Substitution Method
- Topic - Substitution Method by Non-Monic Equations
- Topic - Substitution Method by Equating Equations
- Topic - Substitution Method by Rearranging Equations
- Topic - Substitution Method by Rearranging Non-Monic Equations

#### Numerical Application of Linear Simultaneous Equations

- Video - Number Applications of Simultaneous Equations 1 (2:58)
- Video - Number Applications of Simultaneous Equations 2 (2:54)
- Video - Number Applications of Simultaneous Equations 3 (2:11)
- Video - Number Applications of Simultaneous Equations 4 (1:42)
- Video - Digit Applications of Simultaneous Equations (8:33)

#### Practical Application of Linear Simultaneous Equations

- Video - Coin Value Applications of Simultaneous Equations 1 (3:12)
- Video - Coin Value Applications of Simultaneous Equations 2 (2:52)
- Video - Age Applications of Simultaneous Equations 1 (2:04)
- Video - Age Applications of Simultaneous Equations 2 (3:51)
- Video - Geometry Applications of Simultaneous Equations 1 (3:14)
- Video - Geometry Applications of Simultaneous Equations 2 (3:13)
- Video - Cost Applications of Simultaneous Equations 1 (2:59)
- Video - Cost Applications of Simultaneous Equations 2 (2:01)
- Video - Measurement Applications of Simultaneous Equations (2:23)
- Video - Review Problem Solving of Simultaneous Equations (6:09)

#### Undefined or All Real Values of Simultaneous Equations

- Video - No Points of Intersection of Two Straight Lines (2:16)
- Video - Simultaneous Equations by Substitution Method & Graphs (4:55)
- Video - Simultaneous Equations in No Solutions & Graphs (3:40)
- Video - Theory of Undefined Solutions in Simultaneous Equations (2:56)
- Video - Simultaneous Equations in Undefined Solutions (2:49)
- Video - Simultaneous Equations in All Real Value Solutions (2:20)
- Video - Simultaneous Equations: No Solutions (1:39)
- Video - Simultaneous Equations: All Real values (2:23)
- Video - Intersections of Hyperbola and Linear Equations: Undefined (2:23)

#### Further Graphs of Basic Power Functions

- Video - Graphs of Even Power (4:49)
- Video - Graphs of Odd Power (1:51)
- Video - Graphs of Rational Power 1 by 3 and 1 by 5 (1:35)
- Video - Graphs of Rational Power 2 by 3 and 2 by 5 (3:59)
- Video - Graphs of Rational Power 4 by 3 and 8 by 5 (1:26)
- Video - Graphs of Rational Power 1 by 2 and 1 by 4 (1:30)
- Video - Graphs of Rational Power 3 by 4 and 5 by 6 (1:15)
- Video - Graphs of Rational Power 3 by 2 and 7 by 6 (1:30)

#### INEQUALITIES

- Video - Inequalities in Number Lines (5:22)
- Video - Writing Inequalities from Number Lines (2:29)
- Video - Understanding Linear Inequalities (5:10)
- Video - Simple Inequalities (2:04)
- Video - Linear Inequalities Pronumerals on Both Sides (2:48)
- Video - Reversing Inequality Sign (4:46)
- Video - Two Inequality Signs in Inequalities (5:20)
- Video - Linear Inequalities (4:36)

#### Simultaneous Linear Inequalities

- Video - Common Regions between Horizontal and Straight Lines (3:49)
- Video - Common Regions between Vertical and Horizontal Lines (3:42)
- Video - Common Regions between Straight Lines (1:43)
- Video - Common Regions between Two Straight Lines (4:26)
- Video - Common Regions between Two Straight Lines in General Form (2:15)
- Video - Common Regions between Three Straight Lines (2:54)
- Video - Common Regions involving Vertical and Horizontal Lines (2:21)
- Video - Intersecting Regions involving Linear Inequalities (5:16)

#### Regions of Linear Inequalities

- Video - Understanding Sketching Regions (2:43)
- Video - Regions Bounded by Horizontal Lines (1:41)
- Video - Regions Bounded by Vertical Lines (1:28)
- Video - Regions Bounded by Straight Lines (5:24)
- Video - Regions Bounded by Straight Lines in General Form (6:16)
- Video - Writing Inequalities (5:20)
- Video - Vertical & Horizontal Regions (4:20)
- Video - Regions involving Linear Lines (3:53)