Naming a line by where it crosses the axes

What to assume they know

• One point and a slope, or two points: the same condition written twice — the point-slope and two-point forms and their shared hypothesis
• Lines parallel to an axis, where one coordinate never changes — the axis-parallel families, which will turn out to be the ones the intercept form cannot reach
• Signed coordinates, and that a crossing point on an axis has one coordinate zero
• That division by zero is not a permitted operation, and what it means for a formula's domain
• Solving a pair of simultaneous conditions, one linear and one a product, to recover two unknown intercepts

What they should be able to do

• Derive the slope-intercept equation by applying the point-slope form at the line's y-axis crossing
• Explain what the sign of the y-intercept records
• Derive the x-intercept variant by the same method, and say why the chapter leaves it to the reader
• Derive the intercept form from the two-point form applied to the two axis crossings
• State the two conditions the intercept form requires, and name the three families of lines it excludes
• Convert a linear equation into slope-intercept form and read off slope and y-intercept
• Convert a linear equation into intercept form and read off both intercepts, or explain why the conversion is impossible
• Recover a line from conditions on its intercepts, such as their sum or product
• Use the y-intercepts of two lines as two vertices of a triangle and find its area

Where it usually goes wrong

• "The y-intercept is the distance from the origin to the line." It is a signed coordinate. Its size is a distance; its sign says which side of the origin the crossing falls on. Negative intercepts are ordinary, as Example 7(i) and Example 8 both show.
• "Every line has an intercept form." Three families do not: lines through the origin, and lines parallel to either axis. Exercise 9.3 q2(iii) is one of them, printed inside an exercise that asks for the form.
• "Equal intercepts means x + y = a and nothing else." A line through the origin also has equal intercepts — both zero — and is invisible to the intercept method. Exercise 9.2 q11 is safe only because the required line misses the origin.
• "Slope-intercept form is a different kind of equation from point-slope." It is the point-slope form evaluated at one particular point, chosen because the line supplies it.
• "y = mx + c can express any line." It cannot express a vertical line, because there is no m. That gap is why the general form of Naming Ax + By + C = 0, the one form the distance formula ahead will take is needed.
• "Reducing to intercept form means dividing by the constant term." It means making the right-hand side one and each variable's coefficient into the reciprocal of an intercept. If the constant term is zero the manoeuvre is impossible, which is the same statement as "a line through the origin has no intercept form".
• "An x-intercept is where the line crosses the y-axis." The name records which axis the crossing lies on, not which variable is written. Mixing these up produces answers that are right up to a swap, which is the hardest kind of error to spot.
• "The two intercepts determine the slope, so the intercept form secretly contains it." It does, but only for the lines that have both intercepts. That is exactly why the form is less general than slope-intercept, not more.

Questions to check understanding

• Write a line's equation from its slope and either intercept
• Write a line's equation from its two intercepts, including when one of them is negative
• Reduce a given equation to slope-intercept form and state slope and y-intercept
• Reduce a given equation to intercept form and state both intercepts, or explain why it has none
• Find a line through a given point whose intercepts satisfy a stated sum, product or equality
• Find a line from a condition on the point where the axes cut it, such as a midpoint or a stated ratio
• Find the area of a triangle two of whose sides are given in slope-intercept form and one of which is an axis

Examples worth working on the board

Inputs only. Values marked verified are worked out here on data printed inside pp. 160–175.

• Fig 9.12 (§9.3.4, p. 162). One line rising to the right and crossing the y-axis above the origin, with the crossing point labelled by its coordinate pair — first entry zero, second entry the intercept — and the words "Slope m" printed along the line. Read off the printed page: the crossing point's label sits to the right of the y-axis, and exactly one primed letter appears in the whole drawing — an X with a prime on the negative arm of the horizontal axis. The downward arm of the vertical axis carries no letter at all, so an artist redrawing this figure should not supply a matching prime there.
• The slope-intercept derivation (§9.3.4, p. 162). Verified: the crossing has first coordinate zero, so feeding it into the point-slope form and simplifying leaves the familiar y = mx + c. Nothing is used that One point and a slope, or two points: the same condition written twice did not already establish; the saving is that the point comes free with the line.
• What the sign of c records (§9.3.4, p. 162). Verified: c is positive when the crossing is on the positive half of the y-axis and negative when it is on the negative half. It records position, not distance — the distance is the size of c and carries no sign.
• Case II (§9.3.4, p. 162). If the line's x-intercept is d, the equation is y = m(x − d). Verified: this is the point-slope form applied at (d, 0), and the chapter explicitly leaves the derivation to the reader — so do it. It takes one line.
• Fig 9.13 (§9.3.5, p. 163). One line falling from the y-axis down to the x-axis, with the y-axis crossing labelled as a pair whose first entry is zero and the x-axis crossing labelled as a pair whose second entry is zero. Read off the printed page: the two intercepts are marked as dashed measured lengths, one vertical from the origin up to the y-crossing and one horizontal from the origin across to the x-crossing, and both are drawn positive.
• The intercept-form derivation (§9.3.5, p. 163). Verified: running the two-point form on the two crossings and rearranging produces an equation in which each variable is divided by its own intercept and the two quotients sum to one. The rearrangement divides by both intercepts, which is where the restriction enters.
• The three excluded families. Verified, and the chapter states none of them: a line through the origin has both intercepts zero and no intercept form; a horizontal line other than the x-axis never crosses the x-axis, so has no x-intercept; a vertical line other than the y-axis has no y-intercept. What survives is exactly the lines that cross both axes at distinct points. This is the single most useful thing to say in the whole topic.
• Example 7 (p. 162). Two lines, each with tan θ = 1/2 for the inclination θ; the first has y-intercept −3/2, the second has x-intercept 4. Verified: the first is y = x/2 − 3/2, i.e. 2y − x + 3 = 0; the second is y = (x − 4)/2, i.e. 2y − x + 4 = 0. Two parallel lines whose equations differ only in one constant — an early sighting of the family The gap between two parallel lines as one distance measured once will measure between.
• Example 8 (p. 163). Intercepts −3 on the x-axis and 2 on the y-axis. Verified: the intercept form gives x/(−3) + y/2 = 1, which clears to 2x − 3y + 6 = 0. A negative intercept is admissible; a zero one is not.
• Exercise 9.2 items belonging here, with their data intact:
  • q6: crossing the y-axis 2 units above the origin, at 30° to the positive x-direction. Verified: the slope is 1/√3 and the line is x − √3·y + 2√3 = 0.
  • q11: equal intercepts on both axes, through (2, 3). Verified: writing both intercepts as a gives x + y = a, so a = 5 and the line is x + y = 5. Worth flagging: a line through the origin also has its two intercepts equal, both zero, and no such line has an intercept form — so the phrase "equal intercepts" is quietly assuming they are not zero. The line through the origin and (2, 3) is 3x − 2y = 0, which the intercept method cannot find.
  • q12: through (2, 2), intercepts summing to 9. Verified: the two intercepts satisfy a + b = 9 and 2/a + 2/b = 1, giving a² − 9a + 18 = 0, so the pairs are (3, 6) and (6, 3), and the lines are 2x + y = 6 and x + 2y = 6.
  • q17: P(a, b) is the midpoint of the part of the line between the axes; show the line is x/a + y/b = 2. Verified: the crossings are (2a, 0) and (0, 2b), and the intercept form on those gives the result immediately.
  • q18: R(h, k) cuts the part between the axes in the ratio 1 : 2. Verified: taking the x-axis crossing first, the crossings are (3h/2, 0) and (0, 3k), so the line is 2x/(3h) + y/(3k) = 1. Taking the y-axis crossing first gives a different answer, and the question does not say which end the ratio starts from.
• Exercise 9.3 q1 (p. 167) — reduce to slope-intercept form and give slope and y-intercept. Verified: (i) x + 7y = 0 has slope −1/7 and y-intercept 0; (ii) 6x + 3y − 5 = 0 has slope −2 and y-intercept 5/3; (iii) y = 0 has slope 0 and y-intercept 0.
• Exercise 9.3 q2 (p. 167) — reduce to intercept form and give the intercepts. Verified: (i) 3x + 2y − 12 = 0 gives intercepts 4 and 6; (ii) 4x − 3y = 6 gives 3/2 and −2; (iii) 3y + 2 = 0 is horizontal, has y-intercept −2/3 and no x-intercept at all, so it has no intercept form. Item (iii) is the best thing in this exercise and should be taught as the point rather than marked wrong.
• Miscellaneous Exercise q2 (p. 172) — intercepts whose sum is 1 and whose product is −6. Verified: they are the roots of t² − t − 6 = 0, namely 3 and −2, so the two lines are 2x − 3y = 6 and 3x − 2y + 6 = 0.
• Miscellaneous Exercise q6 (p. 173) — perpendicular to x/4 + y/6 = 1 through the point where that line meets the y-axis. Verified: the meeting point is (0, 6), the given slope is −3/2, so the required line is 2x − 3y + 18 = 0.
• Miscellaneous Exercise q7 (p. 173) — the area of the triangle formed by y − x = 0, x + y = 0 and x − k = 0. Verified: the first two meet at the origin and the third meets them at (k, k) and (k, −k), so the area is k².
• Miscellaneous Exercise q11 (p. 173) — through the crossing of 4x + 7y − 3 = 0 and 2x − 3y + 1 = 0, with equal intercepts. Verified: the crossing is (1/13, 5/13), so the line is 13x + 13y = 6.
• Example 14 (pp. 170–171, Fig 9.18). The triangle cut out by the y-axis together with a line y = m₁x + c₁ paired with a second line y = m₂x + c₂. Verified: two vertices are the two y-axis crossings, at heights c₁ and c₂, so the side along the axis has length |c₁ − c₂|; the third vertex is found by solving the two slope-intercept equations, and its first coordinate is (c₂ − c₁)/(m₁ − m₂), whose size is the triangle's height. Halving the product gives (c₁ − c₂)²/(2|m₁ − m₂|). This is the clearest payoff in the chapter for keeping lines in slope-intercept form: two of the three vertices are read off without any work at all.

Figures to have open

• Fig 9.12 redrawn with the y-axis crossing labelled and the slope written along the line. The chapter's own figure.
• Fig 9.13 redrawn with both crossings labelled and both intercepts drawn as measured dashed lengths. The chapter's own figure, and the measured lengths are what make the form's name intelligible.
• A three-panel schematic of the excluded families — through the origin, horizontal, vertical — each with the offending division shown collapsing. Standard schematic; the chapter provides the cases only inside exercises.
• Example 14's triangle drawn with the two y-axis vertices marked at their intercepts. Standard schematic built from the chapter's own worked example.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 9 "Straight Lines", §9.3.4 Slope-intercept form, pp. 161–162 — Fig 9.12, both cases, and the sign convention for c
• §9.3.5 Intercept - form, p. 163 — Fig 9.13 and the derivation from the two-point form
• Example 7, p. 162, and Example 8, p. 163
• Exercise 9.2, pp. 163–164, questions 6, 11, 12, 17 and 18
• Exercise 9.3, p. 167, questions 1 and 2
• Miscellaneous Exercise on Chapter 9, pp. 172–173, questions 2, 6, 7 and 11
• Example 14, pp. 170–171, and Fig 9.18
• The chapter Summary, p. 175 — both forms and the x-intercept variant restated