Systems of Equations

Lesson 14 of 30: Algebra in Everyday Life

Duration: 45 minutes Class size: 20 students Year Level: 7-10 (students working approximately between Year 2-7 levels)

Learning Objectives (NSW Curriculum Alignment)

By the end of this lesson, students will be able to:

Understand the concept of a system of linear equations and its relevance in real-life scenarios.
Graphically solve systems of two linear equations by plotting and interpreting the point of intersection.
Recognise different types of solutions for systems of equations: one solution, no solution, and infinitely many solutions.
Use tables of values to plot linear equations as a scaffold for graphing systems.
Develop foundational algebraic reasoning skills through visual and contextual methods.

(Aligned with NSW Mathematics K-10 Syllabus: AC9M7N01, AC9M8N05, AC9M9N05—solving linear equations, graphing linear relations and solving systems graphically for Years 7-9) (Achievement Standards included focus on graphing linear relations, solving simultaneous equations graphically by Year 9)

Required Materials:

Graph paper or printed Cartesian grids
Coloured pencils or markers
Whiteboard and markers
Handout with two sets of linear equations for graphing practice
Tablets or laptops with graphing app (optional but recommended for visual learners and repetition)
Dyslexia-friendly worksheet version with clear fonts, large spacing, and colour-coded steps

Lesson Breakdown

1. Introduction & Recap (7 minutes)

Purpose: Connect today’s lesson to prior knowledge and real-life application.

Begin with a simple story: “Imagine you’re buying fruit and have two possible fruit-sellers. Each has a price model for apples and bananas. How can you find when the prices are the same?”
Briefly explain that a system of equations means two equations work together to find where two things meet (intersect).
Review how to graph a single linear equation using tables of values (repeat with simple examples).
Use clear, concise language and visual aids—write examples large on the board using colour to differentiate terms.

2. Guided Activity: Graphing Two Equations (15 minutes)

Purpose: Experience hands-on graphing of two linear equations and find intersection points.

Present a system of two simple linear equations, e.g.: y = 2x + 1 y = -x + 4
Model how to create tables of values for each equation (choose x-values, calculate y-values).
Students work in pairs to complete tables for both equations on their worksheet (dyslexia-friendly version available).
Colour-code each equation’s points to aid visual discrimination.
Students plot the points and draw the two lines on graph paper.
Identify and discuss the point where the two lines intersect, relating this to the solution of the system.
Teacher circulates and supports scaffolding, breaking down steps as needed for students with learning difficulties.

3. Concept Consolidation and Interpretation (8 minutes)

Purpose: Discuss solution types and interpret what they mean.

Show examples of three possibilities using graphs:

One point of intersection (one solution).
Parallel lines (no solution).
Same line overlapping (infinitely many solutions).

Use real-world scenarios for context, e.g.: comparing phone plans or transport options to see if or when they cost the same.
Have students classify a few example graphs or equations.
Encourage students to verbalise what the solution (or lack of) means in context, reinforcing mature communication skills in line with transitioning to workplace expectations.

4. Independent Practice with Support (10 minutes)

Purpose: Practice and reinforce with differentiated support.

Provide additional sets of linear systems with varied difficulty levels:
For emerging learners: simple positive integer slopes and intercepts.
For confident learners: include negative slopes or rearranging equations to y=mx+c form.
Allow students to use graphing apps if available or continue manual plotting depending on their skill and preference.
Circulate, offering personalised assistance, and promoting kindness and respect in peer interactions by pairing students in mixed-ability pairs.
Use guided questioning: “What do you notice about where the lines meet or don’t meet?”, “How do you know this is the solution?”

5. Extension Activities for Advanced Learners (optional)

Challenge advanced students to:
Solve a system algebraically (substitution or elimination method) after graphing.
Investigate what happens to the solution when coefficients change (e.g., use sliders on a graphing tech tool).
Create their own real-life problem involving two quantities and write corresponding systems.

6. Reflection and Assessment (5 minutes)

Purpose: Formative assessment and reflection.

Verbal or written exit question: “Explain how graphing can help us find the solution to two equations.” “Describe a real-life scenario where this skill could be useful.”
Collect worksheets for review.
Provide positive, specific feedback highlighting persistence and effort.
Encourage students to set a small learning goal for next lesson based on today’s work, supporting self-awareness and maturity.

Differentiation Strategies

Use visual supports and colour coding extensively.
Break tasks into small, manageable steps with clear instructions.
Provide repeated modelling and practice with scaffolding.
Pair students strategically to encourage peer support.
Use technology (apps, graphing calculators) for those who benefit from multi-modal input.
Dyslexia-friendly handouts: use sans-serif fonts (e.g., Arial), larger size, significant line spacing, and avoid clutter.
Provide concrete examples and real-world contexts linked to students’ interests or future careers to maintain engagement.

Career and Transition Connection

Briefly discuss how algebra and graphical skills could be used in budgeting, scheduling, or comparing job offers—helping students relate abstract maths to life skills.
Reinforce respectful communication and teamwork during paired activities.
Highlight problem-solving as a valuable skill in workplaces.

This lesson plan seeks to balance NSW syllabus requirements with the needs of diverse learners in a supportive classroom environment, promoting understanding, personal growth, and real-world connection through clear, structured, and engaging teaching methods.

Teaching Instructions