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Introducing Linear Equations in Two Variables
Linear Equations in Two Variables · Problem solving
This unit opens the chapter at the point where linear equations are extended from one unknown to two, anchoring on Radha's fruit purchase as the motivating situation. Watch for students who write, for example, 60x = 50y + 280 — they have mis-assigned the total; insist that each side of the equation represents the same quantity. Students who finish early may read Example 1, p.86, on their own.
0-10Open with Radha's fruit-purchase scenario: x kg of mangoes at ₹60 per kg and y kg of bananas at ₹50 per kg totalling ₹280, writing 60x + 50y = 280 on the board. Ask students what is different about this equation compared to equations they have solved before — there are two unknowns. Elicit from students that fixing one variable (say x = 1) gives a value for the other, and that fixing a different value of x gives a different pair.
10-25Name the object: an equation of the form involving two unknowns to the first degree is a linear equation in two variables. Discuss the two sides of 60x + 50y = 280 — each side represents the total cost, making it a meaningful equation. Have students write two more real-life sentences (e.g., buying pens and pencils, scoring runs in two innings) and express each as a linear equation in two variables. Pairs share and the class critiques whether each is a correct two-variable linear equation.
25-40Work through Exercise Set 13.1 Q3, p.87 together: judge whether Charlie's t = 2p and Meera's p = 2t correctly model the notebook-pen relationship, and whether x + y = 176 correctly models the two-batsmen situation. Students write their verdict and reasoning individually, then discuss as a class which formulation is correct and why.
40-50Consolidate: invite two students to each pose a new situation and write it as a linear equation in two variables; the class checks that the equation is linear (degree 1 in each variable) and involves exactly two unknowns. Summarise on the board: a linear equation in two variables relates two unknown quantities whose values together satisfy one condition.
Competency (C-3.2). Models everyday situations as linear equations in two variables and checks whether a proposed equation correctly represents the condition.
Learning outcome. Students can model a two-condition real-life situation as a linear equation in two variables and verify whether a proposed equation correctly captures the given condition.
Priya buys x notebooks at ₹15 each and y pens at ₹8 each, spending exactly ₹79 in total. Which equation correctly models this situation?
- A 8x + 15y = 79
- B 15x + 8y = 79
- C 15x - 8y = 79
- D 15x + 8y + 79 = 0
B. 15x + 8y = 79
What each wrong option reveals
- A Swaps the price and item variable — assigns ₹8 as the notebook price and ₹15 as the pen price, reversing the roles of x and y.
- C Subtracts the pen cost from the notebook cost instead of adding both costs to form the total — models a difference, not a total expenditure.
- D Moves the total to the left-hand side without changing its sign: 15x + 8y = 79 rearranges to 15x + 8y - 79 = 0, not 15x + 8y + 79 = 0.
Also saved as 10 periods or 13 periods.
- 2Standard Form and Identifying Coefficients
Standard Form of Linear Equations in Two Variables - 3What Counts as a Solution — the Algebraic Meaning
Algebraic Meaning of Solution - 4Plotting Solutions — Every Solution Lies on a Line
Geometric Visualisation of Solutions - 5Justifying Properties of Solutions — True or False Reasoning
Algebraic Meaning of Solution (Revisit) - 6Slope as Steepness — Rise Over Run from Two Points
Slope-intercept form of a Linear Equation in Two Variables - 7Slope-Intercept Form — Deriving and Using y = mx + d
Slope-Intercept Form - 8Modelling Situations as Pairs of Linear Equations
Pair of Linear Equations in Two Variables - 9Method of Substitution — Reducing a Pair to One Variable
Algebraic Methods for Solving a Pair of Linear Equations - 10Method of Elimination — Matching Coefficients and Subtracting
Algebraic Methods for Solving a Pair of Linear Equations - 11How Many Solutions? — Coefficient Ratios and Number of Solutions
Algebraic Methods for Solving a Pair of Linear Equations - 12Solving Pairs Algebraically — Extended Problem Practice
Algebraic Methods for Solving a Pair of Linear Equations - 13Graphical Solution — Intersection as the Common Solution
Graphical Method for Solving A Pair of Linear Equations in Two Variables - 14Parallel and Coincident Lines — Graphical Confirmation of Ratio Rules
Graphical Method for Solving A Pair of Linear Equations in Two Variables - 15Robot Paths, Conversion Graphs and Further Graphical Work
Graphical Method for Solving A Pair of Linear Equations in Two Variables - 16Whole-Chapter Synthesis — Connecting Solutions, Slopes and Systems
Synthesis
- 1. Orienting Yourself: The Use of Coordinates
- 2. Introduction to Linear Polynomials
- 3. The World of Numbers
- 4. Exploring Algebraic Identities
- 5. I'm Up and Down, and Round and Round
- 6. Measuring Space: Perimeter and Area
- 7. The Mathematics of Maybe: Introduction to Probability
- 8. Predicting What Comes Next: Exploring Sequences and Progressions
- 9. Propositions and their Converses
- 10. How Quantities Combine: Understanding Data
- 11. The World of Algorithms
- 12. Quadrilaterals
- 13. Two Variables, One Line
- 14. Math of Space: Surface Area and Volume
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