Recovering y = ax + b from two observations

What to assume they know

• Linear growth and linear decay — linear rules built from a starting value and a per-interval change
• A constant difference is the signature of a linear pattern — that the per-step change is the coefficient
• What makes a polynomial linear, and the equation you get by fixing its value — solving a linear equation in one variable
• Substituting a pair of known values into an expression containing two letters
• Rearranging an equation to express one letter in terms of another
• Substituting one expression into another and simplifying
• Ratios and unit conversion, for the temperature exercise

What they should be able to do

• Explain why a relationship of the form y = ax + b has exactly two unknowns
• Turn each of two observations into an equation in a and b
• Solve the resulting pair by expressing one unknown in terms of the other and substituting
• Interpret the recovered a and b in the language of the original situation
• Obtain a directly as the change in y divided by the change in x, and explain why that shortcut works
• Check a recovered relationship by feeding both original observations back through it
• State the two circumstances in which two observations are not enough
• Set up the relationship when the roles of the two quantities are assigned the other way round, as the temperature exercise does

Where it usually goes wrong

• "One reading is enough — divide the bill by the usage." ₹350 over 10 GB gives ₹35 a GB, which contradicts the second reading. That division silently assumes the fixed fee is zero. Show it failing at 20 GB.
• "a and b are found by trial." They are the solution of two equations that the two observations write down for you. Nothing is guessed.
• "The two observations can be any two." They must sit at different values of x. Two bills for the same 10 GB give one equation twice, and no amount of algebra will separate a from b.
• "If I get numbers out, the model was right." The arithmetic always produces an a and a b. It is the third observation that tests whether the relationship was linear at all. This is why checking a further point matters — and it is what the next topic does with a graph.
• "b is the bigger number, a the smaller." In item 2, a = 60 and b = 200; in item 1, a = 25 and b = 150; in item 3, a is a fraction under one and b is negative. There is no rule of size, only of role.
• "A negative b means something has gone wrong." In item 3 it is correct and meaningful: at 0 °F the Celsius reading is below zero.
• "In °C = a °F + b, a must be 9/5 because that is the formula I know." The exercise has assigned the roles the other way round, so a is 5/9. Read which quantity the equation solves for before reaching for a remembered constant.
• "20 and 150 are just answers." They are the price list. The Think and Reflect box on p. 27 exists to force this reading.

Questions to check understanding

• Given two observations and the form y = ax + b, find a and b
• Interpret the recovered a and b in terms of the situation
• Predict the value of y at an x neither observation supplied
• Set up the relationship when the dependent quantity is the less obvious one, as in the Celsius exercise
• Convert between two measurement scales once the relationship is recovered — the form of End-of-Chapter item 8 on p. 37
• Recover a linear polynomial from two points on its graph, which is End-of-Chapter item 10 on p. 38, stated with the points (1, 5) and (3, 11)

Examples worth working on the board

Inputs below. §2.5 prints no figure. Values marked verified are worked out here; where the chapter prints a value I say so.

• The definition that opens §2.5 (p. 26). The chapter states that a linear relationship holds between two variables x and y and can be written as y = ax + b.
• The tile pattern, rewritten (p. 26). The chapter returns to Fig. 2.4 from p. 21 and says that with x as the term number and y as the tile count, the relationship is y = 2x – 1. This is the same rule as §2.3's 2n – 1; only the letters have changed. Worth a beat, because it is the moment the chapter's two halves are joined.
• Example 11, the data bill (pp. 26–27). A telecom operator bills a flat monthly charge together with a further amount for every GB used. A student's bill was ₹350 for 10 GB, and ₹550 for 20 GB. With x the GB used and y the bill in rupees, and the relationship taken as y = ax + b, find a and b.
• The chapter's working, as printed (pp. 26–27). Substituting gives 350 = 10a + b and 550 = 20a + b. It sets b = 350 – 10a from the first, substitutes into the second to get 550 = 20a + (350 – 10a), simplifies to 550 = 10a + 350, so 10a = 200 and a = 20. Then b = 350 – 10a = 350 – 200 = 150, and the relationship is y = 20x + 150.
• Think and Reflect, p. 27. The box asks what the numbers 20 and 150 stand for. Verified — 20 is the charge per GB and 150 is the fixed monthly fee. The chapter does not print this answer. It is the most important question in the section, because it is where the recovered numbers stop being answers and become facts about the price list.
• The counting argument, which the chapter does not state. Verified, and an added framing — y = ax + b contains two unspecified numbers. Each observation is one linear equation relating them. Two independent equations pin two unknowns; one equation leaves a whole family. Fixing only x = 10, y = 350 admits a = 20, b = 150 and equally a = 35, b = 0 and a = 0, b = 350. Show three such rules all passing through the one observation and diverging at 20 GB — that is what the second observation buys.
• The shortcut. Verified — subtracting 350 = 10a + b from 550 = 20a + b eliminates b at once and gives 200 = 10a, so a = 20 in one line. Then either observation returns b. The chapter uses the substitution route instead; both are correct.
• Why the shortcut is not new. Verified — (550 – 350) ÷ (20 – 10) is the change in y per unit change in x, which is exactly the constant gap of §2.3. The chapter itself makes this identification, but four pages later: on p. 31 it states that the slope of the line for the tile pattern is 2, the same 2 that is the gap in 1, 3, 5, 7. So the connection is the chapter's own, arriving late.
• Checking both ways. Verified — 20 × 10 + 150 = 350 and 20 × 20 + 150 = 550. Both observations must be fed back; checking only one would also pass a = 35, b = 0.
• A prediction, to show what the rule is for. Verified, not in the book — at 15 GB the recovered rule gives ₹450, a value neither observation supplied. Recovering a and b is worth doing only because it lets you answer questions you never measured.
• Exercise Set 2.5 (p. 27), three items, all inputs to hand over:
  • A learning platform billing a flat monthly charge plus an amount for each module accessed. 10 modules billed ₹400; 14 modules billed ₹500. With y the bill and x the module count in y = ax + b, find a and b.
  • A gym billing a flat monthly charge plus an amount for each hour of badminton court use. 10 hours billed ₹800; 15 hours billed ₹1,100. With y the bill and x the hours in y = ax + b, find a and b.
  • The relationship between Celsius and Fahrenheit, given as °C = a °F + b. Ice melts at 0 °C and 32 °F; water boils at 100 °C and 212 °F. Find a and b. The item's printed hint restates the two pairings.
• The exercise answers, as checks. Verified — item 1 gives a = 25 and b = 150, so y = 25x + 150. Item 2 gives a = 60 and b = 200, so y = 60x + 200. Item 3 gives a = 5/9 and b = –160/9.
• Item 3 is the interesting one. Verified — the exercise puts Celsius on the left, so Celsius is the dependent quantity and the coefficient comes out as the fraction 5/9 rather than the familiar 9/5. Note also that item 1's inputs 10 and 14 differ by 4, and item 2's 10 and 15 by 5, so neither divides as neatly as Example 11's 10; students who have only seen the worked example will reach for a difference of ten. Cross-reference within the chapter: End-of-Chapter item 8 on p. 37 gives a Kelvin-to-Fahrenheit relationship in the other direction, as y = (9/5)(x – 273) + 32, so the two items between them show the same physical relationship written both ways round.

Figures to have open

• A panel that can display several candidate rules through one fixed observation and show them separating at a second x. Standard schematic; this is the argument of sections 3 and 4 and it is not in the book.
• A two-equation solving layout that can show the substitution route and the subtraction route side by side. Standard schematic.
• A price-list card for the recovered rule — a per-GB rate and a fixed fee — so a and b land back in the situation. Standard schematic.
• No graph is needed in this topic; the graphical treatment begins on p. 27 with §2.6 and belongs to Why two points are enough to draw the line.
• No artwork from the textbook is required; §2.5 prints none.

Where this sits in the book

• NCERT Ganita Manjari, Class 9, printed Chapter 2, "Introduction to Linear Polynomials", §2.5 "Linear Relationships", pp. 26–27. The definition and the tile-pattern restatement are on p. 26; Example 11 begins on p. 26 and finishes at the top of p. 27; Exercise Set 2.5 occupies the middle of p. 27.
• The Think and Reflect box on p. 27 asks what the recovered numbers mean and is the hinge of this topic.
• Backward links inside the chapter: the tile pattern and its rule are on pp. 21–22, covered in A constant difference is the signature of a linear pattern; the constant term as a fixed charge appears in the chess club example on p. 19.
• Forward links inside the chapter: a is named the slope on p. 31 and b the y-intercept on p. 35, both in What a and b do to the line: slope, y-intercept, and parallel families. The identification of the slope with the constant gap of a sequence is stated on p. 31.
• End-of-Chapter Exercises: starred item 8 on p. 37 (Kelvin and Fahrenheit), starred item 10 on p. 38 (recover a linear polynomial from two points), starred items 11 and 13 on pp. 38–39 (two linear polynomials constrained by several conditions at once).