Why a pair of these equations is a pair of straight lines

What to assume they 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

What they should be able to do

• Turn a stated situation into two equations by naming the two unknown counts and writing one equation per condition
• Explain why the set of points satisfying a linear equation in two variables is a straight line, treating the vertical case separately
• Produce a two-point table for a given equation and justify why two points are enough to fix the line
• State what a solution of a pair is, and check a candidate pair against both equations rather than one
• Identify the common point of two drawn lines as the solution of the pair
• Distinguish the points of the line that answer the equation from the points that also make sense in the story the equation came from

Where it usually goes wrong

• "An equation has an answer, so this one has an answer." A single linear equation in two unknowns has infinitely many, one for every point of a line. Nothing has gone wrong; the question was simply under-determined until the second condition arrived.
• "The graph is a drawing of the equation." It is the equation's solutions, every one of them, laid out. Ask the class to test a point they pick off the line, and a point just beside it.
• "y = ½x and x – 2y = 0 are different equations, so different lines." Multiplying through by 2 and moving terms changes the writing, not the set of pairs that satisfy it. This is the seed of the whole coincident-lines case later in the chapter.
• "Plot lots of points to be safe." Two correct points already determine the line. A third is a check on arithmetic, not on the mathematics — which is why the chapter's tables stop at two.
• "Any point on the line is a possible answer for Akhila." The line runs through fractional and negative coordinates, and she cannot take minus two rides or half a ride. The line is the solution set of the equation; the story admits only some of its points.
• "You could just keep trying numbers." Trial closed here because the answer was a small whole number. Give the class a pair whose answer is a fraction and the method has nowhere to go. That extension is added here: §3.3 opens by making the same complaint about graphing, not about trying values, when it says a crossing at non-integral coordinates is easy to misread. The parallel is worth drawing, but it should be drawn rather than credited to the algebraic half of the chapter.

Questions to check understanding

• Form a pair of equations from a stated situation by naming the two unknowns
• Given a pair, produce two solutions of each equation and plot both lines
• Decide whether a stated pair of values satisfies a given pair of equations
• Read the solution of a pair off a drawn graph and verify it algebraically
• Explain why a linear equation in two variables has more than one solution
• Say which points of a drawn line are admissible answers to a stated word problem and why the rest are not

Examples worth working on the board

Inputs, in the chapter's own quantities. Values marked verified are worked out here on those inputs; the chapter does not print them here.

• Akhila at the fair (§3.1, p.24). She takes rides on a Giant Wheel and plays hoopla — the stall game where a thrown ring keeps whatever it fully covers. She plays hoopla half as many times as she rides the wheel. A ride costs 3, a hoopla game 4, and she spends ` 20 in total. The page asks how many of each. A framed panel of artwork sits under the text, centred and running to roughly two-thirds of the text width rather than the full measure, and drawn in pen and ink with a blue wash — not a photograph. It shows stalls, a crowd, a ring being thrown at objects on a counter, and the wheel at the right edge.
• The trial route the page floats. It suggests testing one ride, then two, and so on.verified, one ride costs 3 and half a ride's worth of hoopla is not a whole game at all; two rides cost 6 and one hoopla game 4, total 10; three rides give 9 plus one and a half games, again not whole; four rides give 12 plus two games at 8, total 20. The route works, but only because the numbers were kind and the answer happened to be small.
• The two conditions written out (§3.1, p.25). With x the number of rides and y the number of hoopla games, the chapter sets down y = ½x for the halving condition and 3x + 4y = 20 for the money. It then says several ways of solving await in the chapter, and solves nothing on the page.
• Carrying it through. Verified: substituting ½x for y in the money condition gives 3x + 2x = 20, so x = 4 and y = 2. Four rides at 3 is 12, two hoopla games at 4 is 8, and 12 + 8 = ` 20. This value is an added derivation, not something the chapter states in §3.1 — the explanation may present it as the answer the chapter is heading towards, not as a result it prints.
• The same pair in the general form (§3.2, p.26, Table 3.1, row 1). The chapter's first tabulated pair is x – 2y = 0 together with 3x + 4y – 20 = 0, which is Akhila's pair with the halving condition cleared of its fraction, and the table records it as a crossing pair with one solution. Verified: y = ½x and x – 2y = 0 are satisfied by exactly the same pairs, and the crossing point is (4, 2). Closing the explanation on this row is the strongest possible ending — the story's answer and the table's classification are the same fact.
• Two-point tables. The chapter's plotting tables always carry exactly two columns of values (Table 3.2, p.27; Table 3.3, p.28). For x – 2y = 0, x = 0 gives y = 0 and x = 4 gives y = 2; for 3x + 4y = 20, x = 0 gives y = 5 and x = 4 gives y = 2. Verified. Both tables contain (4, 2), which is the point the two lines share.

Figures to have open

• The chapter's fair artwork (§3.1, p.24) sets the scene and is the only photograph-like image in the topic. Redraw it as a simple schematic — a wheel, a hoopla counter, a price tag on each — rather than reproducing the printed drawing.
• A single coordinate grid carrying both of Akhila's lines with their crossing marked at (4, 2). This is not printed in §3.1; the chapter states the equations and leaves the picture to the reader. Standard schematic.
• A strip diagram of the plane thinning to a line as one condition is imposed, then to a point as the second is imposed. Standard schematic, and the visual carrier of the thesis.
• Row 1 of Table 3.1 (§3.2, p.26), redrawn as a clean four-cell strip.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 3 "Pair of Linear Equations in Two Variables", §3.1 Introduction, pp. 24–25; the two equations for the fair situation are set on p. 25.
• The opening of §3.2 (p. 25) and Table 3.1 (p. 26) supply the general form and the first tabulated pair, which is the fair problem restated.
• Tables 3.2 and 3.3 (pp. 27–28) are the chapter's model for a two-point plotting table.
• Backward pointer: the chapter names Class IX as where linear equations in two variables were met (p. 24).