- 1. What a Linear Equation Really Tells Us
Substitute values into y = mx + c to generate coordinate pairs, complete a table of values for a linear equation, identify the gradient (m) and y-intercept (c), and state what the gradient and y-intercept each represent on a graph.
- 2. When a Line Has Only One Direction
Write equations and draw graphs for horizontal and vertical lines, and use the satisfaction test to check if a point lies on a given line.
- 3. Unwrapping the Equation — Rearranging to y = mx + c
Learn to rearrange equations like ax + y = c and y ± bx = c into the standard y = mx + c form to easily read the gradient and y-intercept.
- 4. Unwrapping the Equation — Rearranging to y = mx + c
Learn to rearrange equations like ax + y = c and y ± bx = c into the standard y = mx + c form to easily read the gradient and y-intercept.
- 5. Lines in Disguise — Plotting ax + by = c
Generate coordinate pairs for an implicit equation, complete a table of values, plot the graph, and learn the intercept method.
- 6. Lines in Disguise — Plotting ax + by = c
Generate coordinate pairs for an implicit equation, complete a table of values, plot the graph, and learn the intercept method.
- 7. How Steep Is the Line? — Gradient by Direction and Measurement
Classify line steepness visually, measure gradients using the slope triangle method, and understand rise over run.
- 8. Gradient Without a Graph — The Formula
Calculate steepness algebraically from coordinate pairs instead of reading from a drawn graph.
- 9. From Graph to Equation — Deriving y = mx + c
Learn the three-step method to read the equation of a line directly from its graph or from two coordinate points.
- 10. One Equation, Two Unknowns — Infinitely Many Solutions
Explain why a single linear equation in two variables has infinitely many solutions, generate valid solution pairs, and see why a second constraint is needed to determine one unique pair.
- 11. Adding and Subtracting Equations — Zero Pairs and Identical Pairs
Add or subtract two linear equations to form a third valid equation; recognise zero pairs and identical pairs; choose addition or subtraction to eliminate one variable; write the resulting one-variable equation without solving it yet.
- 12. The Full Elimination Algorithm — Five Steps
Execute the complete five-step elimination algorithm for systems where coefficients already match; use back-substitution to find the second variable; write the solution as an ordered pair; verify in both original equations.
- 13. Scaling One Equation Before Eliminating
Identify which variable to target for elimination when coefficients do not already match; determine the single multiplier needed to force matching coefficients for one variable; scale one equation correctly by multiplying every term on both sides; execute the complete five-step elimination algorithm after applying a scaling step.
- 14. Scaling Both Equations — Choosing the Efficient Path
Students learn to identify when both equations require scaling because no variable coefficients match, calculate the LCM for both variables, and choose the most efficient elimination path.
- 15. The Substitution Method — Rearrange, Substitute, Solve
Learn to solve simultaneous equations by isolating one variable, substituting into the second equation, and solving systematically.
- 16. Choosing Between Elimination and Substitution
Students learn to examine a pair of simultaneous equations and choose between elimination or substitution based on structural efficiency, executing either method fluently and verifying the solution.
- 17. From Words to Equations — Setting Up the Model
Students learn to translate contextual word problems into systems of two simultaneous equations by defining unknowns, translating conditions, solving, and verifying in context.
- 18. More Complex Word Problems — Ages, Perimeters, and Rates
Students learn to translate non-direct relationships such as age timelines, geometric constraints, and fixed-plus-variable cost structures into algebraic systems, solving and verifying contextually.
