Why a graph stops being trustworthy once the answer is not a whole number

What to assume they know

• Crossing, parallel or lying on top: what each picture says about solutions — solving a pair by drawing both lines and taking the crossing
• Predicting which of the three you will get by comparing coefficients — predicting the case from the coefficients
• Surds as numbers with a definite position on a number line, and decimal approximations to them
• Comparing fractions with unlike denominators, and converting them to decimals
• Reading a scale, and the idea that a reading is only as fine as the marks

What they should be able to do

• State what the graphical method delivers well and what it delivers only approximately
• Estimate where an irrational or awkward-fractional coordinate falls on ruled paper, and say how finely it can be read
• Explain why the limit is the sheet's rather than the reader's — at classroom scale the drawing itself cannot hold the distinction the answer needs
• Show that a pair with a whole-number answer is a special case, not the normal one
• Recognise a pair whose lines are so close together that a hand drawing cannot separate them
• Say what an exact method must supply that a drawing cannot, and why the chapter turns to algebra at this point

Where it usually goes wrong

• "The graphical method gives wrong answers." It gives the right answer to whatever accuracy the drawing supports. The crossing is where it is; the loss happens when a human reads a position off paper.
• "Use bigger graph paper and it will work." Enlarging helps until the next awkward number. A crossing at √3 has no finite decimal to land on at any scale.
• "Since graphs are unreliable, drop them." The drawing is why anyone believes there are exactly three cases, and it is still the fastest way to see what a pair means. The chapter keeps it and adds to it.
• "Nearly parallel lines are parallel." Two lines of slightly different steepness cross exactly once, possibly far outside the sheet. Only the coefficients settle that.
• "An answer with a square root in it means I made a mistake." The chapter puts a surd coordinate up as a normal possibility. The awkwardness is in reporting it from a drawing, not in its existence.
• "Rounding the reading is close enough." Substitute a rounded reading back into both equations and neither will balance. The check the chapter keeps demanding after every worked example is exactly what catches this.

Questions to check understanding

• Say whether a given pair is a sensible candidate for the graphical method, with a reason
• Solve a pair graphically and then check the reading by substitution
• Given a crossing point in fractions, state what the drawing could and could not have told you
• Explain why an exact method is needed, referring to a specific pair
• Identify from the coefficients that two lines are parallel but close, and say what a drawing would have suggested instead

Examples worth working on the board

Inputs. Values marked verified are worked out here; the chapter prints the coordinates but does none of this estimating.

• The chapter's own turn (§3.3 opening, p. 30). It says the graphical route is inconvenient once the crossing has coordinates that are not whole numbers, offers three of them as warnings, notes how easily such readings go wrong, and asks whether another route exists before announcing that several algebraic ones do.
• The three warning coordinates, exactly as printed: (√3, 2√7), (–1.75, 3.3) and (4/13, 1/19). Verified as decimals: √3 is about 1.732 and 2√7 about 5.292; 4/13 is about 0.3077 and 1/19 about 0.0526.
• What that means on paper. Take the common classroom setting of one unit to a centimetre on paper ruled in 2 mm squares, so one small square is a fifth of a unit and a careful eye resolves about half of one — a tenth of a unit. Verified, and the numbers are the whole argument: √3 and 1.7 differ by 0.032 of a unit, which at this scale is 0.32 mm — a third of the tenth-of-a-unit the eye was just granted, so on a hand-drawn sheet the two are simply the same mark. That is the point, and it is stronger than a near miss would be. As for the other pair, 4/13 is 3.1 mm out, so it lands in the second small square rather than the first; 1/19 is 0.53 mm out, about half a millimetre, which is inside the first square and again below what the eye can place. No sharper pencil rescues that; the sheet has no finer mark to read against.
• Everything graphed in the first half landed on whole numbers. Verified, worked from the printed data: Example 1 gives (6, 0); Example 3 gives (1, 0); Exercise 3.1 Q1(i) gives three boys and seven girls; Q1(ii) gives 3 for a pencil and 5 for a pen; Q4(iii) gives (2, 2); Q5 gives 20 m by 16 m; Q7 has vertices (–1, 0), (4, 0) and (2, 3). Every one is a whole-number point. That is a property of the questions chosen, not of pairs of equations.
• The very next worked pair breaks the pattern (§3.3.1, Example 4, p. 30): 7x – 15y = 2 together with x + 2y = 3, whose answer the chapter reaches as x = 49/29 and y = 19/29. Verified: 7(49/29) – 15(19/29) = (343 – 285)/29 = 2, and 49/29 + 2(19/29) = 87/29 = 3, so the pair really is answered there. Verified as decimals: about (1.690, 0.655) — a point no reading off ruled paper would give you as twenty-ninths. Use this as the payoff of the topic even though the substitution work itself belongs to the next brief.
• Two lines a drawing cannot separate (Exercise 3.1 Q4(iv), p. 29): 2x – 2y – 2 = 0 with 4x – 4y – 5 = 0. Verified: these reduce to x – y = 1 and x – y = 1.25, two parallel lines whose perpendicular separation is 0.25/√2, about 0.18 of a unit — under 2 mm at one centimetre to the unit. Drawn by hand they merge, and the honest graphical verdict would be the wrong one. The coefficient test of the previous topic settles it in one line, and so does elimination.
• A gentler contrast for the same point (Exercise 3.1 Q4(ii), p. 29): x – y = 8 with 3x – 3y = 16, which reduce to x – y = 8 and x – y = 16/3. Verified: separation (8 – 16/3)/√2, about 1.89 units, wide enough to see. The two questions differ only in how far apart the lines sit, which is the argument in one comparison.

Figures to have open

• A ruled grid zooming in on an irrational coordinate until the ruling itself is the limit. Standard schematic, and the central image of the topic; the chapter prints nothing like it.
• The first two small squares out from the origin, magnified, with 1/19 marked about half a millimetre from the axis and 4/13 marked just inside the second square. Do not draw both inside one square — 4/13 is 3.1 mm out at one centimetre to the unit, and the second square begins at 2 mm. Standard schematic.
• Two panels at one scale: Exercise 3.1 Q4(iv) with its two lines almost touching, and Q4(ii) with its two lines clearly apart. Both must be drawn to the same unit or the comparison collapses.
• A restatement panel listing the three warning coordinates from p. 30 in the chapter's own order.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 3, §3.3, p. 30 — the paragraph opening the algebraic half, with its three sample coordinates, and the announcement of the methods that follow.
• Example 4, §3.3.1, p. 30, supplies the fractional answer used in section 8; the method itself belongs to Substitution: rewriting one unknown so only the other survives.
• Exercise 3.1 Q4, p. 29, supplies the two close-parallel pairs.
• Backward pointers within the chapter: Example 1 and Fig. 3.1 (pp. 26–27) and Example 3 with Fig. 3.2 (pp. 27–28) are the graphed solutions this topic looks back at.