Predicting which of the three you will get by comparing coefficients

What to assume they know

• Crossing, parallel or lying on top: what each picture says about solutions — the three mutual positions of two lines and the solution count attached to each
• Writing an equation in the general form, with every term on one side
• Equivalent fractions, and comparing two ratios by cross multiplication
• Handling negative numbers in a ratio, and knowing that division by zero is not defined
• Rearranging a linear equation to isolate y

What they should be able to do

• 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
• Form the three ratios and compare them
• Predict crossing, parallel or coincident lines from the comparison, without drawing anything
• Explain why equality of the first two ratios is a statement about steepness
• Explain the part the constants play once the first two ratios agree
• Distinguish the implication the chapter derives from its table from the converse it then asserts, and say which one a prediction actually uses
• Construct a second equation that makes a given equation into a crossing, a parallel or a coincident pair

Where it usually goes wrong

• "There are three rules here." There is one question — is the second equation a multiple of the first? — asked at two depths. Left-hand side only, or all the way through.
• "You compare a1 with b1." The pairing is a with a and b with b, across the two equations. Comparing within one equation answers a different question. The cross product a1b2 = a2b1 is the same statement written without the grouping, and showing that once settles the confusion for good.
• "The constants never matter." They decide the entire difference between two lines lying on each other and two that never meet. They are only irrelevant in the crossing case, where the first comparison has already settled it.
• "Every pair fits one of the three ratio patterns." A third ratio can fail to exist — write a pair whose second constant is zero while the first is not, and c1/c2 is undefined. The comparison must then be made as a cross product, or by scaling one equation and looking. This case appears in the exercise set in form, though not in Exercise 3.1 — every pair there has both constants non-zero, so every third ratio in that set is defined. The nearest printed instance is Exercise 3.2 Q1(v) on p. 33, where both constants are zero and the third ratio reads 0/0, which the recipe also has no rule for.
• "If I write the equations the other way round I get a different answer." Swapping which equation is first turns each ratio into its reciprocal. Equalities stay equalities and inequalities stay inequalities, so the verdict is untouched.
• "The table proves the test." The table shows three worked cases and yields the implication in one direction. Prediction runs the other way, and the chapter asserts that direction rather than deriving it. That is worth saying out loud rather than hiding.
• "Signs can be dropped when forming ratios." Do not reach for Table 3.1's first row to show this — its ratios are 1/3 and −1/2, and stripping the minus leaves 1/2, still unequal to 1/3, so the verdict survives the vandalism intact. The chapter does hand you a case where the sign carries the whole answer: Exercise 3.1 Q3(iii) on p. 29, whose ratios are 1/6 and −1/6. As printed those differ, so the lines cross and the pair is consistent with one solution. Drop the minus and they match, and since the constant ratio is 1/2 and matches neither, the recipe now reports parallel lines and no solution at all. One character, and the answer inverts.

Questions to check understanding

• Compare the ratios for a given pair and state whether the lines cross, are parallel or coincide
• Compare the ratios and state whether the pair is consistent or inconsistent
• Given one equation, write a second producing each of the three cases
• Find the value of a parameter for which a pair has no solution, or infinitely many — the standard board extension of Q6
• Clear fractional or decimal coefficients before applying the test
• Justify a classification in words, not merely assert it

Examples worth working on the board

Inputs. Values marked verified are worked out here on the chapter's data.

• Table 3.1 (§3.2, p. 26) is the topic's central figure. Three rows, each a pair already reduced to the general form, with separate cells for a1/a2, b1/b2 and c1/c2, then the comparison, then the picture, then the reading in terms of solutions.
  • Row 1: x – 2y = 0 with 3x + 4y – 20 = 0. Ratio cells 1/3, –2/4, 0/–20.
  • Row 2: 2x + 3y – 9 = 0 with 4x + 6y – 18 = 0. Ratio cells 2/4, 3/6, –9/–18.
  • Row 3: x + 2y – 4 = 0 with 2x + 4y – 12 = 0. Ratio cells 1/2, 2/4, –4/–12. Verified: row 1 has 1/3 against –1/2, unequal, so the lines cross; row 2 has 1/2 throughout, so one line twice; row 3 has 1/2 for the first two and 1/3 for the third, so two lines that never meet.
• The cross-product reading. Verified, and the argument of sections 4 and 5: for row 3, a1b2 = 1 × 4 = 4 and a2b1 = 2 × 2 = 4, equal — so both equations place y against x at the same rate, and rearranged they read y = –x/2 + 2 and y = –x/2 + 3. Same steepness, different height. For row 1, a1b2 = 1 × 4 = 4 against a2b1 = 3 × (–2) = –6, unequal, and the rearranged forms y = x/2 and y = –3x/4 + 5 do have different steepness.
• What the chapter asserts after the table (§3.2, p. 26). It draws the three implications out of the table — crossing gives unequal first two ratios, coincident gives all three equal, parallel gives the first two equal and the third different — and then states that the reverse implications hold as well for any pair of lines, inviting the reader to test that on further examples of their own choosing. It offers no proof of the reverse direction.
• Exercise 3.1 Q2 (p. 29), three pairs to classify as crossing, parallel or coincident: 5x – 4y + 8 = 0 with 7x + 6y – 9 = 0; 9x + 3y + 12 = 0 with 18x + 6y + 24 = 0; 6x – 3y + 10 = 0 with 2x – y + 9 = 0. Verified: 5/7 against –2/3 gives a crossing; 1/2 in all three cells gives one line; 3 and 3 and 10/9 gives parallel lines.
• Exercise 3.1 Q3 (p. 29), five pairs to call consistent or inconsistent: 3x + 2y = 5 with 2x – 3y = 7; 2x – 3y = 8 with 4x – 6y = 9; (3/2)x + (5/3)y = 7 with 9x – 10y = 14; 5x – 3y = 11 with –10x + 6y = –22; (4/3)x + 2y = 8 with 2x + 3y = 12. Verified: consistent with one answer; inconsistent; consistent with one answer, since (3/2)/9 is 1/6 while (5/3)/(–10) is –1/6; dependent and so consistent, all three ratios being –1/2; dependent and so consistent, all three ratios being 2/3. Two of these five carry fractional coefficients, and the whole difficulty of the question is clearing them correctly before any ratio is formed.
• Exercise 3.1 Q6 (p. 29) is the constructive form and the best classroom moment in the section: given 2x + 3y – 8 = 0, write a second equation making the pair cross, then one making the lines parallel, then one making them coincide. Verified as sample answers the explanation may build live: any equation whose x and y coefficients are not in the ratio 2 : 3, say 3x + 2y – 8 = 0, gives a crossing; 4x + 6y – 5 = 0 keeps the ratio 1/2 on the left but not on the right, so the lines never meet; 4x + 6y – 16 = 0 doubles the whole equation, so it is the same line. Every parallel answer and every coincident answer is a rescaling, which is the thesis restated as a construction.
• The summary's misprint (§3.4, p. 37). Point 4 of the chapter summary lists the three cases again, and its first line prints the second ratio with the same subscript on top and bottom rather than the two different ones. The intended statement is the one on p. 26. Say so if a student raises it; do not build the explanation's statement of the criterion from p. 37.

Figures to have open

• Table 3.1 (§3.2, p. 26) redrawn cleanly, with room to show one row at a time. The chapter's own figure and the anchor of the topic.
• A two-panel schematic pairing each ratio pattern with its picture, built so the panel can be filled in from either side — ratios first, or picture first.
• A steepness diagram: one unit right, and the drop or rise that –a/b gives, drawn for both equations of a pair on the same axes. Standard schematic; the chapter does not draw this and it is what makes section 5 land.
• A constructed family: one fixed line, with a rescaled copy and a shifted copy drawn against it, for Exercise 3.1 Q6.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 3, §3.2, pp. 25–26: the three sample pairs on p. 25, Table 3.1 on p. 26, and the three implications with the statement about the reverse direction immediately below the table.
• Exercise 3.1, p. 29, questions 2, 3 and 6.
• §3.4, p. 37, point 4 restates the criterion; see the note above about its misprinted first line.
• Deliberate cross-reference outside this chapter's pages: the book's Appendix 1, Proofs in Mathematics, pp. 218–238, is where converses and what it takes to establish one are discussed. Chapter 3 does not point there itself, so use it as teacher background rather than presenting it as a link the chapter makes.