3. Pair of Linear Equations in Two Variables

6 topics1 h

Chapter 3 of NCERT Mathematics for Class 10: Pair of Linear Equations in Two Variables. Two sections — Two equations drawn as two lines and Solving without drawing. Six videos, 1 h in all.

Worked answers to the 12 exercise questions in this chapter: Exercise 3.1 · Exercise 3.2 · Exercise 3.3

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Part 1

Two equations drawn as two lines

13 minWhy a pair of these equations is a pair of straight linesOne equation in two unknowns does not pin them down - it thins the whole plane to a line, and that line is not a picture of the equation, it is the complete list of pairs that satisfy it. So two equations are two lists, and the values that answer both at once are exactly the points the two lines have in common.13 min · 14 parts15 minCrossing, parallel or lying on top: what each picture says about solutionsTwo distinct points determine one and only one line. That single fact decides the whole topic: if two lines share two points they share every point, so a pair of linear equations has no solution, exactly one, or endlessly many - and never exactly two. The three pictures are not three cases to memorise. They are the only three the plane permits.15 min · 15 parts15 minPredicting which of the three you will get by comparing coefficientsComparing a1/a2 with b1/b2 is not a third rule to learn beside the three pictures - it is the pictures, arithmetic-side out. Those two ratios agree exactly when the two lines lean the same way, and two lines that lean the same way cannot cross, which leaves the constants one thing to settle. Three patterns, one question, asked at two depths.15 min · 14 parts

Part 2

Solving without drawing

12 minWhy a graph stops being trustworthy once the answer is not a whole numberA graph is never wrong. It is only ever READ - and a reading is a measurement, carrying a resolution the mathematics does not have. Root three sits 0.32 mm from 1.7 on centimetre-ruled paper, which is a third of what a careful eye can place, and it lands on no ruling mark at any fineness whatsoever. Keep the picture for the meaning. Take the number from arithmetic.12 min · 13 parts14 minSubstitution: rewriting one unknown so only the other survivesAn equation is not a fact about a letter. It is PERMISSION TO REPLACE it - and because the swap runs both ways, the pair you end up solving has exactly the same answers as the pair you were given. That is the whole of substitution; everything else is arithmetic. Measured over 565 pairs and 1109262 points, the two systems agreed every single time.14 min · 13 parts16 minElimination: scaling the equations so a variable cancels on additionAdding two true equations gives a third true one, and scaling either by a number leaves its solutions untouched — elimination's only two permissions, and scaling by nought breaks the method outright.16 min · 13 parts

What you will be able to do

  • Turn a stated situation into two equations by naming the two unknown counts and writing one equation per condition
  • Argue from "two points determine a line" that a linear pair cannot have exactly two solutions
  • Put a given pair into the general form and read off a1, b1, c1 from the first equation and then a2, b2, c2 from the second, signs included
  • State what the graphical method delivers well and what it delivers only approximately
  • Choose which equation to rearrange and which letter to isolate, and justify the choice by the arithmetic it avoids
  • Justify multiplying an equation by a non-zero constant, and adding one equation to another, in terms of the solutions each step preserves

What this chapter assumes you already know

  • Linear equations in two variables from Class IX, and the fact that such an equation has many solutions rather than one
  • Plotting a point from its coordinates, and the four quadrants
  • Substituting a number for a letter and checking whether an equation holds
  • Solving a linear equation in one unknown
  • Reading a word problem and naming the unknown quantities
  • Plotting points and drawing a line through two of them
  • That an equation multiplied through by a non-zero number has the same solutions
  • Reading coordinates off a grid, including negative ones

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "An equation has an answer, so this one has an answer."
  • "The graph is a drawing of the equation."
  • "y = ½x and x – 2y = 0 are different equations, so different lines."
  • "Plot lots of points to be safe."
  • "Any point on the line is a possible answer for Akhila."
  • "You could just keep trying numbers."
  • "Two lines could cross twice."
  • "Parallel means they get closer but never touch."