Class 10 Maths Chapter 3 PPT (20 Slides) – Pair of Linear Equations in Two Variables | USP Indore

Class 10 Maths Chapter 3 PPT—Pair of Linear Equations in Two Variables | 20 Slides | USP Indore

Download the complete 20-slide-style PowerPoint presentation for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables, prepared by Unique Study Point (USP), Indore. This PPT is based on the CBSE NCERT syllabus (Session 2026-27) and covers every concept, method, and word problem type of the chapter in a highly visual and student-friendly format.

Complete Topic Coverage — 20 Slides

📘 Slide 1 — Chapter Overview (Topic List)

• Introduction – What is a pair of linear equations?
• Graphical Method of Solution
• Conditions for Consistency and Inconsistency
• Algebraic Methods: Substitution Method
• Algebraic Methods: Elimination Method
• Finding the Value of k
• Word Problems – Age, Speed, Coins, Fractions, Profit, Digits, Pipes

📗 Slide 2 — Introduction: What is a Pair of Linear Equations?

• General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
• Condition: a₁² + b₁² ≠ 0
• NCERT context: Akhila at Hoopla fair — forming equations from real-life situations (3x + 4y = 20 and y = x/2)
• Three possibilities: One solution, No solution, Infinite solutions (coincident)

📐 Slide 3 — Graphical Method: Solve and Find the Triangle

• Draw the graph of 2x + y = 4 and 2x − y = 4
• Find the vertices A, B, C of the triangle formed by these lines and the y-axis
• Find the area of this triangle

📊 Slide 4 — Conditions for Consistency and Inconsistency

• Unique Solution (Consistent): a₁/a₂ ≠ b₁/b₂ → Lines intersect at ONE point (e.g., x + y = 4 and x − y = 2 → solutio(3, 1)) ))
• No Solution (Inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → Parallel lines (e.g., 2x + 3y = 6 and 4x + 6y = 15)
• Infinite Solutions (Consistent/Dependent): a₁/a₂ = b₁/b₂ = c₁/c₂ → Coincident lines (e.g., 2x + y = 4 and 4x + 2y = 8)

🔄 Slide 5 — Substitution Method (4 Steps)

• Step 1: Express one variable in terms of the other from either equation
• Step 2: Substitute this value into the OTHER equation
• Step 3: Solve the resulting single-variable equation
• Step 4: Back-substitute to find the other variable
• ⁣ ractice problem: 3x + 2y = 11 and 2x − y = 2

✂️ Slide 6 — Elimination Method (4 Steps)

• Step 1: Multiply equations to make coefficients of one variable EQUAL
• Step 2: Add or Subtract the equations to ELIMINATE one variable
• Step 3: Solve the resulting single-variable equation
• Step 4: Substitute back to find the other variable
• Practice problem: 5x − 3y = 11 and 4x + y = 7

🔍 Slide 7 — Finding the Value of k

• Q1: For what value of k do kx + 3y = k − 3 and 12x + ky = k have no solution? (Use: a₁/a₂ = b₁/b₂ ≠ c₁/c₂)
• Q2: Finthe valueue of k for which 3x − y + 8 = 0 and 6x − ky = −16 are coincident lines. (Use: a₁/a₂ = b₁/b₂ = c₁/c₂)
• Q3: For what value of k will 2x + ky = 1 and 3x − 5y = 7 have a unique solution? (Use: a₁/a₂ ≠ b₁/b₂)

🌍 Slide 8 — Real-Life Applications of Linear Equations

• Shopping (Kirana Store): 2x + 3y = 220
• Transport and Taxi Fare: a + 10b = 105
• Savings and Banking (8% and 9% interest): 8x + 9y = 186000
• Sports and Tickets (cricket match): x + y = 1000
• Speed and Distance (boat upstream/downstream): 30/(x−y) + 28/(x+y) = 7
• Pipes and Work: 1/x + 1/y = 1/12

🎂 Slide 9 — Word Problems: Age

• Q1: Father\'s age is six times his son\'s age. Four years hence, Father will be four times Son\'s age. Find present ages.
• Q2: Two years ago, Salim was thrice as old as his daughter. Six years later, he will be four years older than twice her age. Find their present ages.
• Doodle Tip: Lethe son\'s\'s agbe x and thex, father\'age be y;y; form TWO equations, then solve.

🔢 Slide 10 — Word Problems: Numbers and Digits

• Q1: Thesumm of a two-digit number and its reverse is 66. Digits differ by 2. Find the number.
• Q2: Thesumm of digits of a two-digit number is 9. Nine times the number equals twice the number with reversed digits. Find the number.
• Q3: Two numbers are ithe ratioio 5:6. If 8 is subtracted from each, the ratio becomes 4: 5. Find the numbers.
• Doodle Tip: Two-digit number = 10x + y; Reversed = 10y + x

💰 Slide 11 — Word Problems: Coins and Money

• ⁣⁣ : Aruna has Re 1 and Rs 2 coins. Total coins = 50; total amount = Rs 75. Find the number of each type.
• Q2: Meena withdrew Rs 2000 frothe banknk using Rs 50 and Rs 100 notes only. She got 25 notes. Find the number of each.
• Q3: Cost of 4 pens and 4 pencil boxes = Rs 100. Three times the cost of a pen is Rs 15 more than a pencil box. Find the cost of each.
• Doodle Tip: LeRsRe 1 coins x andx, Rs 2 coins = y; x + y = total coins; x + 2y = total amount

🚢 Slide 12—Word Problems: Speed, Distance and Time

• Q1: A motorboat travels 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.
• Q2: A person rowing at 5 km/h in still water takes thrice as long going 40 km upstream as going 40 km downstream. Find the speed of the stream.
• Q3: Ankita travels 14 km home partly by rickshaw and partly by bus. Takes 30 minutes if she travels 2 km by rickshaw; takes 9 minutes longer if she travels 4 km by rickshaw. Find the speed of each.
• Doodle Tip: Upstream speed = (boat − stream); Downstream speed = (boat + stream)

📐 Slide 13 — Word Problems: Fractions and Angles

• Q1: A fraction becomes 9/11 if 2 is added to both the numerator and denominator. If 3 is added to both, it becomes 5/6. Find the fraction.
• Q2: Angles of a triangle arx, y, y and 40°. The difference between x and y is 30°. Find x and y.
• Q3: Theangless of cyclic quadrilateral ABCare e: ∠A = (6x+10)°, ∠B = (5x) °, ∠C = (x+y) °, ∠D = (3y−10) ° Find x, y, and all four angles.
• Doodle Tip: Sum of angles of triangle = 180°; Sum of opposite angles of cyclic quadrilateral = 180°

💹 Slide 14 — Word Problems: Profit and Pipes

• Q1: Jamila sold a table and chair for Rs 1050, making 10% profit on table and 25% on chair. Had she made 25% on the table and 10% on the chair, she would have gotten Rs 1065. Find the cost price of each.
• Q2: Two pipes together fill a pool in 12 hours. Larger pipe for 4 hours + smaller for 9 hours = only half full. Find time each pipe takes alone.
• Q3: A shopkeeper sells a saree at 8% profit and sweater at 10% discount → Rs 100 8. At 10% profit on a saree and 8% discount on a sweater → Rs 1028. Find the cost price of the saree and the list price of the sweater.
• Formula: SP = CP × (100 + profit%) / 100

📱 Slide 15 — Word Problems: Mobile Plans (Break-even)

• Plan A: Fixed Rs 99/month + Rs 2 per extra minute
• Plan B: Fixed Rs 199/month + Rs 1 per extra minute
• At how many minutes will both plans cost the SAME? (Break-evepoint.) t)
• Equations: Plan A: y = 99 + 2x; Plan B: y = 199 + x; Solve: 99 + 2x = 199 + x

🚢 Slide 16 — Solved Example: Motorboat (Upstream/Downstream)

• Full step-by-step solution: 30/(x−y) + 28/(x+y) = 7 and 21/(x−y) + 21/(x+y) = 5
• Substitution: u = 1/(x−y), v = 1/(x+y) → Convert to standard form → Solve → Find x and y

🚰 Slide 17 — Solved Example: Two Pipes Problem

• Full solution: Let large pipe = hrs and thes, small pipe = y hrs
• Equations: 1/x + 1/y = 1/12 and 4/x + 9/y = 1/2
• Substituting u = 1/x, v = 1/y → Equation 1: u + v = 1/12 → Equation 2: 4u + 9v = 1/2

🍽️ Slide 18 — Solved Example: Restaurant Thali and Auto Fare

• Restaurant: 3 thalis + 2 cold drinks = Rs 240; 5 thalis + 3 cold drinks = Rs 390. Find the price of 1 thali and 1 cold drink.
• Saree and Sweater profit/discount probl—fullll equation setup

🛺 Slide 19 — Solved Example: Auto Rickshaw Fare

• 10 km journey → paid Rs 105; 15 km journey → paid Rs 155
• Fixed charge = Ra;a, charge per km = Rs b
• Equation 1: a + 10b = 105; Equation 2: a + 15b = 155
• Find a and b, then find cost for 25 km journey

⭐ Slide 20 — Quick Revision Mind Map

• General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
• Consistency Table: all three ratio conditions with graph type
• Substitution Method3-stepps summary
• Elimination Method3-stepps summary
• Value of k: using ratio conditions
• Word Problems: Age | Coins | Speed | Fractions | Profit | Digits | Pipes

Key Features of This PPT

• ✅ 20 slides — complete NCERT Chapter 3 coverage
• ⁣ All solving methods: Graphical, Substitution, Elimination
• ⁣ All word problem types covered with doodle visuals
• ✅ Finding value of k — all 3 cases (unique, no solution, coincident)
• ⁣ Full step-by-step solved examples for boat, pipes, auto fare
• ⁣ Quick Revision Mind Map on final slide
• ⁣ Prepared as per CBSE NCERT Class 10 Maths Syllabus 2026-27

Important Formulas Covered

• General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
• Unique solution: a₁/a₂ ≠ b₁/b₂
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
• Upstream speed = (boat speed − stream speed)
• Downstream speed = (boat speed + stream speed)
• SP = CP × (100 + profit%) / 100
• Two-digit number = 10x + y; Reversed = 10y + x

Important FAQs — Class 10 Maths Chapter 3

Q. What are the three methods to solve a pair of linear equations?
Ans. The three methods are: (1) Graphical Method – Plot both equations on a graph and find the point of intersection. (2) Substitution Method – Express one variable in terms of the other and substitute. (3) Elimination Method – Multiply equations to make coefficients equal, then add or subtract to eliminate one variable.

Q. What is the condition for a pair of linear equations to have no solution?
Ans. When a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the pair of equations has no solution. The lines are parallel and never intersect. Example: 2x + 3y = 6 and 4x + 6y = 15.

Q. What is the condition for infinite solutions in a pair of linear equations?
Ans. When a₁/a₂ = b₁/b₂ = c₁/c₂, the pair of equations has infinitely many solutions. The lines are coincident (overlap completely). Example: 2x + y = 4 and 4x + 2y = 8.

Q. How do you find the value of k in a pair of linear equations?
Ans. Use the ratio conditions. For no solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. For coincident lines: a₁/a₂ = b₁/b₂ = c₁/c ₂. For unique solution: a₁/a₂ ≠ b₁/b₂. Set up the ratio equation with k and solve.

Q. How to solve boat and stream problems using linear equations?
Ans. Let boat speed in still water = x km/h and stream speed = y km/h. Then, downstream speed = x + y and upstream speed = x − y. Use Time = Distance/Speed to form two equations, then solve using the substitution u = 1/(x − y) and v = 1/(x + y) to simplify.

Q. How are two-pipe problems solved using linear equations?
Ans. Let the large pipe fill the tank in x hours and the small pipe in y hours. Then, in one hour, the large pipe fills 1/x of the tank, and the small one fills 1/y. Set up equations based on given conditions. Substitute u = 1/x and v = 1/y to convert to a simple pair of linear equations.

About Unique Study Point (USP), Indore

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