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CBSE Class 10 · Maths · board exam 2027

Chapter 3: Pair of Linear Equations in Two Variables — notes

The chapter’s 6 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.

This chapter’s important questions →

  1. 1. Linear equations in two variables

    Look at this equation on the board.

    • • 2x + 3y = 5
    • • two variables: x and y, power of each = 1
    • • General form: ax + by + c = 0
    • • a, b, c are real numbers; a and b not both 0
    • • x = 1, y = 1:
    • • LHS = 2(1) + 3(1) = 2 + 3 = 5 = RHS ✓
    • • x = 1, y = 7:
    • • LHS = 2(1) + 3(7) = 2 + 21 = 23 ≠ 5 ✗
    • • All the solutions together make a straight line
    • • A pair of linear equations:
    • • a₁x + b₁y + c₁ = 0
    • • a₂x + b₂y + c₂ = 0
    • • Akhila: x = number of rides, y = number of Hoopla games
    • • y = 1/2x … (1)
    • • 3x + 4y = 20 … (2)
    • • y = 1/2x
    • • 2y = x
    • • x − 2y = 0

  2. 2. Solving a pair by graph

    Every solution of a linear equation is a point, and all its points lie on one straight line.

    • • Each equation → a straight line
    • • Solution of the pair = the point common to both lines
    • • Line 1: x − 2y = 0 ⇒ x = 2y
    • • | y | x = 2y | point |
    • • | 0 | 2(0) = 0 | (0, 0) |
    • • | 1 | 2(1) = 2 | (2, 1) |
    • • Line 2: 3x + 4y = 20 ⇒ y = (20 − 3x)/4
    • • | x | y | point |
    • • | 0 | (20 − 0)/4 = 5 | (0, 5) |
    • • | 4 | (20 − 12)/4 = 2 | (4, 2) |
    • • Check (4, 2) in line 1: 4 − 2(2) = 4 − 4 = 0 ✓
    • • Lines meet at (4, 2) ⇒ x = 4, y = 2
    • • ① Intersecting lines: one common point → unique solution
    • • 2x + 3y = 9 and 4x + 6y = 18
    • • 2 × (2x + 3y = 9) gives 4x + 6y = 18: the same line
    • • x = 3, y = 1: 2(3) + 3(1) = 6 + 3 = 9 ✓
    • • ② Coincident lines: every point common → infinitely many solutions
    • • x + 2y − 4 = 0 and 2x + 4y − 12 = 0
    • • second ÷ 2: x + 2y − 6 = 0
    • • x + 2y = 4 and x + 2y = 6 together: impossible
    • • ③ Parallel lines: no common point → no solution
    • • Has a solution → consistent
    • • No solution (parallel) → inconsistent
    • • Coincident lines → dependent (and consistent)

  3. 3. Comparing the ratios of coefficients

    Drawing a graph takes time.

    • • a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0
    • • Compare a₁/a₂, b₁/b₂, c₁/c₂
    • • | Ratios | Lines | Solutions | Pair |
    • • | a₁/a₂ ≠ b₁/b₂ | intersecting | exactly one | consistent |
    • • | a₁/a₂ = b₁/b₂ = c₁/c₂ | coincident | infinitely many | dependent, consistent |
    • • | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | parallel | none | inconsistent |
    • • Akhila: x − 2y = 0, 3x + 4y − 20 = 0
    • • a₁ = 1, b₁ = −2; a₂ = 3, b₂ = 4
    • • a₁/a₂ = 1/3, b₁/b₂ = (−2)/4 = −1/2
    • • 1/3 ≠ −1/2 ⇒ intersecting, one solution
    • • Romila: 2x + 3y − 9 = 0, 4x + 6y − 18 = 0
    • • 2/4 = 1/2, 3/6 = 1/2, (−9)/(−18) = 1/2
    • • all three equal ⇒ coincident, infinitely many solutions
    • • Rails: x + 2y − 4 = 0, 2x + 4y − 12 = 0
    • • 1/2 = 2/4 = 1/2, (−4)/(−12) = 1/3
    • • 1/2 = 1/2 ≠ 1/3 ⇒ parallel, no solution
    • • Find k: 3x + 2y = 5, 6x + ky = 10 (infinitely many solutions)
    • • a₁/a₂ = b₁/b₂ = c₁/c₂
    • • 3/6 = 1/2, 5/10 = 1/2
    • • 2/k = 1/2 ⇒ k × 1 = 2 × 2 ⇒ k = 4
    • • Check: 2 × (3x + 2y = 5) gives 6x + 4y = 10 ✓

  4. 4. Substitution method

    A graph is good for seeing, but when the answer is a fraction like 19 upon 29, no graph can show it exactly.

    • • Step 1: from one equation, write one variable in terms of the other
    • • Step 2: substitute it in the other equation and solve
    • • Step 3: put this value back to get the other variable
    • • 7x − 15y = 2 … (1)
    • • x + 2y = 3 … (2)
    • • From (2): x = 3 − 2y … (3)
    • • 7(3 − 2y) − 15y = 2
    • • 21 − 14y − 15y = 2
    • • −29y = 2 − 21
    • • −29y = −19
    • • y = (−19)/(−29) = 19/29
    • • x = 3 − 2(19/29) = 3 − 38/29
    • • = 87/29 − 38/29 = 49/29
    • • Check (1): 7(49/29) − 15(19/29) = (343 − 285)/29 = 58/29 = 2 ✓
    • • x = 49/29, y = 19/29
    • • x + 2y − 4 = 0 ⇒ x = 4 − 2y
    • • 2(4 − 2y) + 4y − 12 = 0
    • • 8 − 4y + 4y − 12 = 0
    • • −4 = 0 ✗ false statement
    • • False statement ⇒ no solution (parallel lines)
    • • x = (9 − 3y)/2
    • • 4((9 − 3y)/2) + 6y = 18
    • • 18 − 6y + 6y = 18
    • • 18 = 18 ✓ always true
    • • Always true ⇒ infinitely many solutions (coincident lines)

  5. 5. Elimination method

    The second algebraic method is elimination, and it is often the fastest.

    • • Step 1: multiply so that one variable has the same coefficient in both
    • • Step 2: add or subtract to remove that variable
    • • Step 3: solve, then put the value back
    • • Incomes 9x, 7x; expenditures 4y, 3y (in rupees)
    • • Saving = income − expenditure
    • • 9x − 4y = 2000 … (1)
    • • 7x − 3y = 2000 … (2)
    • • (1) × 3: 27x − 12y = 6000 … (3)
    • • (2) × 4: 28x − 12y = 8000 … (4)
    • • (4) − (3): 28x − 27x − 12y + 12y = 8000 − 6000
    • • x = 2000
    • • 9(2000) − 4y = 2000
    • • 4y = 18000 − 2000 = 16000
    • • y = 16000/4 = 4000
    • • Incomes: 9 × 2000 = 18000, 7 × 2000 = 14000
    • • Check: 18000 − 4(4000) = 18000 − 16000 = 2000 ✓
    • • 14000 − 3(4000) = 14000 − 12000 = 2000 ✓
    • • Monthly incomes: ₹ 18000 and ₹ 14000
    • • 2x + 3y = 8 … (1), 4x + 6y = 7 … (2)
    • • (1) × 2: 4x + 6y = 16 … (3)
    • • (3) − (2): 0 = 9 ✗ false statement
    • • False ⇒ no solution (inconsistent)

  6. 6. Word problems: from words to equations

    Most board questions come as a story.

    • • ① Let x = …, y = …
    • • ② Two sentences → two equations
    • • ③ Solve (substitution or elimination)
    • • ④ Check in the words of the question
    • • ⑤ Answer with the unit
    • • Sum of two numbers 70, difference 10
    • • Let larger = x, smaller = y
    • • x + y = 70 … (1)
    • • x − y = 10 … (2)
    • • (1) + (2): 2x = 70 + 10 = 80
    • • x = 80/2 = 40
    • • 40 + y = 70 ⇒ y = 70 − 40 = 30
    • • Check: 40 + 30 = 70 ✓, 40 − 30 = 10 ✓
    • • The numbers are 40 and 30
    • • 47 = 10 × 4 + 7
    • • tens digit x, units digit y ⇒ number = 10x + y
    • • digits reversed ⇒ number = 10y + x
    • • age now = x
    • • 5 years ago = x − 5
    • • 10 years later = x + 10

Summary

Very good! You have learnt a lot today, so let us revise it together once. A linear equation in two variables is a x plus b y plus c equals zero, and its graph is a straight line. Two such equations make a pair. The lines intersect, are coincident, or are parallel, which gives one solution, infinitely many, or none. Then we compared the ratios a-one upon a-two, b-one upon b-two and c-one upon c-two to know the type without a graph. We solved pairs by substitution and by elimination, and turned story questions into equations. Write the ratio table in your notebook once without looking. Today's homework is on your screen. In the next chapter we will meet quadratic equations, where x comes with a square. If anything was not clear, press the Ask button and ask me. Done, you worked very hard today!

AskMedha · Medha Ma’am — CBSE Class 10 Maths, Chapter 3. Written from the textbook, checked twice.

Questions parents and students ask

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Where are these notes written from?
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