Crossing, parallel or lying on top: what each picture says about solutions

What to assume they know

• Why a pair of these equations is a pair of straight lines — that each equation of the pair is a line, and that a solution of the pair is a point on both
• 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

What they should be able to do

• Argue from "two points determine a line" that a linear pair cannot have exactly two solutions
• Name the three mutual positions of two lines in a plane and give the solution count that goes with each
• Use consistent, inconsistent and dependent correctly, and say why a dependent pair is a consistent one
• Build a two-point table for each equation of a pair, plot both lines, and read the crossing
• Verify a solution read off a graph by substituting it into both equations
• Recognise a pair whose two equations are the same equation in disguise, by scaling one of them
• Say what a drawn picture has settled and what it has only made plausible

Where it usually goes wrong

• "Two lines could cross twice." They cannot, and the reason is the one the whole topic turns on: two shared points force the lines to be the same line. Draw two lines and try to make them meet twice — the attempt is the argument.
• "Parallel means they get closer but never touch." Parallel lines keep a fixed separation. Nothing dramatic happens far off the page; there is simply no crossing anywhere.
• "Coincident lines are two lines lying on each other." They are one line with two names. That is why the chapter reaches for equivalent when it defines the dependent case.
• "Dependent and consistent are opposites." They overlap by definition: a dependent pair has endlessly many solutions, so it certainly has one, so it is consistent. The chapter says so explicitly on p. 25.
• "Inconsistent means the equations are wrong." Both equations can be perfectly good statements. Inconsistent says only that nothing satisfies both at once — as in the rails of Example 7 later in the chapter.
• "An answer of zero means no answer." Champa buys no skirts, and y = 0 is the answer, not the absence of one. The empty case and the zero case are different things.
• "If the lines look parallel on my sheet, they are." A drawing shows the span you happened to plot. Lines with nearly equal steepness cross far outside the page — which is why the chapter supplies a coefficient test in the same section, and why Example 2 is settled by scaling rather than by drawing.

Questions to check understanding

• Given a pair, classify it as consistent, inconsistent or dependent and justify the label
• Solve a consistent pair graphically and state the coordinates of the crossing
• Given a graph of two lines, state how many solutions the pair has
• Explain why a pair of linear equations cannot have exactly two solutions
• Show that two given equations are equivalent by scaling one of them
• Word problems whose answer is read off a drawn graph, including cases where the answer is zero
• Find the vertices and area of the triangle two lines cut with an axis

Examples worth working on the board

Inputs, in the chapter's own quantities. Values marked verified are worked out here; the chapter prints its own verdicts where noted.

• The three sample pairs (§3.2, p. 25). Printed with their verdicts attached, in this order: x – 2y = 0 with 3x + 4y – 20 = 0, which cross; 2x + 3y – 9 = 0 with 4x + 6y – 18 = 0, which coincide; x + 2y – 4 = 0 with 2x + 4y – 12 = 0, which run parallel. Verified: doubling the first equation of the second pair gives the second equation of that pair exactly, so they are one line; doubling the first equation of the third pair gives 2x + 4y – 8 = 0, which disagrees with 2x + 4y – 12 = 0 in its constant alone, so the two lines can never meet.
• Table 3.1 (§3.2, p. 26). Eight columns wide: serial number, the pair, then a1/a2, b1/b2, c1/c2 as separate cells, then the comparison of the ratios, then the graphical picture, then the algebraic reading. Its three rows are the three pairs above, with ratio cells 1/3, –2/4, 0/–20 for row 1; 2/4, 3/6, –9/–18 for row 2; and 1/2, 2/4, –4/–12 for row 3. The right-hand cells read as crossing lines with exactly one answer, coincident lines with endlessly many, and parallel lines with none.
• Example 1 (§3.2, pp. 26–27): x + 3y = 6 together with 2x – 3y = 12, asked as a consistency question. Table 3.2 supplies the plotting values, written as y = (6 – x)/3 taking x = 0 and x = 6, and y = (2x – 12)/3 taking x = 0 and x = 3. The four points are A(0, 2), B(6, 0), P(0, –4) and Q(3, –2), and the chapter states that the lines have B in common, so x = 6 with y = 0 answers the pair. Verified: 6 + 3(0) = 6 and 2(6) – 3(0) = 12, so B satisfies both. Note: B is the crossing point and one of the four tabulated points, which is a coincidence of this example, not a general feature.
• Fig. 3.1 (§3.2, p. 27). A square ruled grid, axes lettered X′X across and Y′Y up. The numerals stop well short of the drawn axes: along the horizontal they run −1 to 5, with nothing at 6 even though that is where the lines meet — B's position is given only by its point label — and along the vertical they run −4 up to 1, with no 2 and no 3 anywhere. The axes themselves are drawn past their last numeral in every direction, to roughly 7 across and 3 up, so a redraw must not take the drawn extent for the numbering. Both lines are drawn with arrowheads at each end and each carries its own equation lettered along it inside the artwork; A, B, P and Q are labelled with their coordinates at the points. The crossing sits on the x-axis at the right of the frame.
• Example 2 (§3.2, p. 27): 5x – 8y + 1 = 0 together with 3x – (24/5)y + 3/5 = 0. The chapter scales the second by 5/3 and gets the first back, concluding that the lines coincide and the answers are endless, and it leaves the plotting to the reader. Verified: (5/3)(3x) = 5x, (5/3)(–24/5)y = –8y, (5/3)(3/5) = 1, so the scaled equation is the first, character for character. Verified by the ratio route as well: 5 ÷ 3, –8 ÷ (–24/5) and 1 ÷ (3/5) are all 5/3.
• Example 3 (§3.2, pp. 27–28), Champa at a sale. She reports her skirt count twice over: it falls two below twice her pants count, and also four below four times that same count. With x pants and y skirts the chapter writes y = 2x – 2 and y = 4x – 4. Table 3.3 takes x = 2 and x = 0 for the first, giving y = 2 and y = –2, and x = 0 and x = 1 for the second, giving y = –4 and y = 0. The lines cross at (1, 0), so she bought one pair of pants and no skirt. Verified: 2(1) – 2 = 0 and 4(1) – 4 = 0.
• Fig. 3.2 (§3.2, p. 28). The same style of ruled grid, axes lettered X′X and Y′Y, with the two lines rising steeply and their equations lettered along them inside the artwork. Four labelled points: A(2, 2), B(0, –2), Q(1, 0) and P(0, –4). The crossing is at Q, on the x-axis.
• Exercise 3.1 material worth showing (pp. 28–29). Q1(i): ten pupils in a quiz, girls exceeding boys by four. Q1(ii): five pencils with seven pens cost 50, and seven pencils with five pens cost 46. Q4 asks which of four pairs are consistent and to solve the consistent ones graphically: x + y = 5 with 2x + 2y = 10; x – y = 8 with 3x – 3y = 16; 2x + y – 6 = 0 with 4x – 2y – 4 = 0; 2x – 2y – 2 = 0 with 4x – 4y – 5 = 0. Q5: a rectangular garden whose half-perimeter measures 36 m, its length beating its width by 4 m. Q7: draw x – y + 1 = 0 and 3x + 2y – 12 = 0, find the triangle they cut with the x-axis and shade it. Verified, as inputs the explanation may check itself against: Q1(i) gives three boys and seven girls; Q1(ii) gives 3 a pencil and 5 a pen; Q4 runs coincident, parallel, crossing at (2, 2), parallel; Q5 gives 20 m by 16 m; Q7 has vertices (–1, 0), (4, 0) and (2, 3).

Figures to have open

• Fig. 3.1 (§3.2, p. 27) and Fig. 3.2 (§3.2, p. 28) both carry their equations as lettering inside the artwork, so a redrawn schematic must reproduce that labelling or the picture loses the link back to the algebra. Redraw rather than copy; the coordinates and the crossing are the content.
• A three-panel schematic of crossing, parallel and coincident lines, drawn to one scale so the panels can be compared. Standard schematic, and the spine of the topic.
• A movement showing two lines pinned at one shared point and then forced through a second — collapsing onto each other. Standard schematic; this is the proof of the thesis and nothing in the chapter draws it.
• Table 3.1 (§3.2, p. 26) redrawn as three clean rows. The ratio cells need to be legible because the next topic reads them closely.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 3, §3.2, whose printed heading reads "Graphical Method of Solution of a Pair of Linear Equations", pp. 25–29. Definitions of the three words, p. 25; the three sample pairs, p. 25; Table 3.1, p. 26.
• Example 1 with Table 3.2 and Fig. 3.1, pp. 26–27. Example 2, p. 27. Example 3 with Table 3.3 and Fig. 3.2, pp. 27–28.
• Exercise 3.1, pp. 28–29, questions 1, 4, 5 and 7.
• Forward pointer inside the chapter: §3.4's summary, p. 37, restates the three cases in the order intersect, coincide, parallel — matching Table 3.1 and the chapter's three sample pairs, but not the prose enumeration on p. 25, which runs intersect, parallel, coincident. This brief's own sections 3, 4 and 5 follow the prose order. The chapter prints both, so say which is meant whenever an ordering is claimed.