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CBSE Class 9 · Mathematics

Introduction to Linear Polynomials

Official NCERT chapter from Ganita Manjari Part I (book code iemh1). ExamMaster notes are original teaching at CBSE Class 9 depth.

This lesson follows the official chapter “Introduction to Linear Polynomials” in Ganita Manjari Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 9
  • Medium level
  • 8 concepts

1Linear Polynomials

A linear polynomial in x is a writing ax + b with a ≠ 0 (degree 1). 3x − 5 is linear; 3x² − 5 is not. The graph of y = ax + b is a straight line.

A “line-like” curve that bends is not this polynomial.

Figure. A linear polynomial is degree 1, so its graph is a straight line. p(x)=2x+1 through (0,1) and (3,7) on the inset axes (origin at 0.08, 0.92). Checked: 7=2·3+1.

How it works

  1. See the highest power of xMust be 1.
  2. See a nonzero coefficient on that xa ≠ 0.
  3. Call it linear; refuse a square termThe sort.

2Exploring linear patterns

A linear pattern in a table is a constant change in y when x steps by 1: +3 each time is slope 3. If the change drifts, the pattern is not linear.

Two points always sit on some line; a whole table must keep the same step.

Figure. A linear pattern adds the same step each time. Marks 3, 7, 11, 15 sit at those values on a 0–16 line (3/16, 7/16, 11/16, 15/16 of the span). Each gap is +4.

How it works

  1. Read Δy for each +1 in xThe step.
  2. See if that step staysLinear if yes.
  3. Write y = (step)x + start if the start is namedThe pattern.

3Linear Relationships

A linear relationship is y = mx + c: m is the rate, c is the value at x = 0. Two different m’s are two different tilts. Parallel lines share m and differ in c.

c is not the “middle” of the line.

Figure. A linear relationship is a constant add: y = x+3. The marked point (11,14) is the pair that makes x+3=14 true. Axes origin is the inset L; the series does not claim to pass through it.

How it works

  1. Read m as rise over runThe rate.
  2. Read c as the y when x is 0The intercept.
  3. Write y = mx + cThe relationship.

m and c from a table

When x = 0, y = 4. Each time x grows by 1, y grows by 3. Write y.

  • c4
  • m3
  • y3x + 4

Pro tip. Constant step is the slope. The start at x=0 is c.

4Visualising linear relationships

Visualising means a straight line on the plane: two points determine it; a third point is a check. A scatter that weaves is not this visualise.

A thick marker that hides a bend is a lying line.

Figure. Visualising x+3=9: start at the unknown 6, hop +3, land on 9. Marks sit at 6/12 and 9/12 of a 0–12 line. The hop is the given addend, not a guess.

How it works

  1. Plot two points from the ruleThe line’s spine.
  2. Join them straightThe picture.
  3. Test a third pair from the ruleIt must sit on the join.

5A definition is a test you can run

Linear polynomial is a test: degree 1, a ≠ 0. If you cannot name the degree, you have a writing only.

A pretty 3x−5 banner is not the test.

Figure. A definition is a test you can run: substitute. x=7 makes x+3=10 true; x=8 gives 11, so it fails. The letter is the number that passes, not a decoration.

How it works

  1. Name the highest powerThe object.
  2. See it is 1 and a ≠ 0The test.
  3. Then the word has contentThe definition ran.

6Name the given before the unknown

The given is the table or the writing. The unknown is m, c, or “is it linear?”. Copy the steps in y before you invent a slope.

Using only the first and last y and ignoring a middle drift is a silent given.

Figure. In x+8=16, name the givens before the unknown: the addend 8 and the total 16 are known. Only then ask which number sits in x (here 8).

How it works

  1. Copy the pairs or the expressionThe given.
  2. Name m, c, or the sortThe unknown.
  3. Then compute the stepGiven first.

7One worked case is enough at this class

One line or one table is enough. A page of 3x+4 clones does not add a new degree-1 rule.

Ten twins of the same tilt are still one idea.

Figure. One worked case is enough: x+6=16, undo the +6, x=10. Check: 10+6=16. Do not pile extra examples that teach the same undo.

How it works

  1. Write one y = mx + c or test one tableThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

8A miss: swapping the school name for the picture

A rising-road photo is a setting. The maths is y = mx + c. A slope you “feel” from the photo without a run is the swap-miss.

A hill selfie is not m = 3.

Figure. The miss is swapping the school name for the picture: treating x as a pretty letter instead of the number that makes the sentence true. For x+2=7 that number is 5.

How it works

  1. Keep the road as a storyA setting.
  2. Write the rate and the interceptThe maths.
  3. Plot or computeDo not swap.
3x² − 5 is
  1. Not linear — degree 2
  2. Linear
  3. A constant

Linear means degree 1.

Notes

  • The official chapter title is “Introduction to Linear Polynomials”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • linear: y = mx + c
  • degree of ax+b (a≠0) is 1

Recap

Hold these pegs from the official chapter “Introduction to Linear Polynomials”. The wording is ExamMaster’s teaching, not a textbook recap.

Linear Polynomials
A linear polynomial in x is a writing ax + b with a ≠ 0 (degree 1).
Exploring linear patterns
A linear pattern in a table is a constant change in y when x steps by 1: +3 each time is slope 3.
Linear Relationships
A linear relationship is y = mx + c: m is the rate, c is the value at x = 0.
Visualising linear relationships
Visualising means a straight line on the plane: two points determine it; a third point is a check.
A definition is a test you can run
Linear polynomial is a test: degree 1, a ≠ 0.
Name the given before the unknown
The given is the table or the writing. The unknown is m, c, or “is it linear?”. Copy the steps in y before you invent a slope.

Practise Introduction to Linear Polynomials

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