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Pair of Linear Equations in Two Variables

Karnataka Board · Class 10 · Mathematics

Flashcards for Pair of Linear Equations in Two Variables — Karnataka Board Class 10 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.

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20 Flashcards
Card 1Substitution Method

Solve by substitution method: x + y = 10 and 2x - y = 5

Answer

Step 1: From first equation: y = 10 - x Step 2: Substitute in second equation: 2x - (10 - x) = 5 Step 3: Simplify: 2x - 10 + x = 5 → 3x = 15 → x = 5 Step 4: Find y: y = 10 - 5 = 5 Answer: x = 5, y = 5…

Card 2Elimination Method

Solve by elimination method: 3x + 2y = 12 and 2x + 3y = 13

Answer

Step 1: Make coefficients equal. Multiply first by 3, second by 2: 9x + 6y = 36 ... (1) 4x + 6y = 26 ... (2) Step 2: Subtract (2) from (1): 5x = 10 → x = 2 Step 3: Substitute x = 2 in original: 3(2) +…

Card 3Graphical Analysis

Find the values of a₁/a₂, b₁/b₂, c₁/c₂ for: 2x + 3y - 6 = 0 and 4x + 6y - 12 = 0. What type of lines are these?

Answer

Given: 2x + 3y - 6 = 0 and 4x + 6y - 12 = 0 a₁ = 2, b₁ = 3, c₁ = -6 a₂ = 4, b₂ = 6, c₂ = -12 Step 1: Calculate ratios: a₁/a₂ = 2/4 = 1/2 b₁/b₂ = 3/6 = 1/2 c₁/c₂ = -6/-12 = 1/2 Step 2: Compare: a₁/a₂…

Card 4Word Problems

A shopkeeper has ₹50 and ₹100 notes totaling ₹4000. If he has 50 notes in total, find the number of each type of note.

Answer

Let x = number of ₹50 notes, y = number of ₹100 notes Step 1: Form equations: Total notes: x + y = 50 ... (1) Total value: 50x + 100y = 4000 ... (2) Step 2: From (1): x = 50 - y Step 3: Substitute i…

Card 5System Analysis

Check if the system has unique solution, no solution, or infinitely many solutions: x + 2y = 5 and 2x + 4y = 8

Answer

Given: x + 2y - 5 = 0 and 2x + 4y - 8 = 0 a₁ = 1, b₁ = 2, c₁ = -5 a₂ = 2, b₂ = 4, c₂ = -8 Step 1: Calculate ratios: a₁/a₂ = 1/2 b₁/b₂ = 2/4 = 1/2 c₁/c₂ = -5/-8 = 5/8 Step 2: Compare: a₁/a₂ = b₁/b₂ ≠…

Card 6Special Cases

Solve: 2/x + 3/y = 13 and 5/x - 4/y = -2 (where x ≠ 0, y ≠ 0)

Answer

Let u = 1/x and v = 1/y Transformed equations: 2u + 3v = 13 ... (1) 5u - 4v = -2 ... (2) Step 1: Multiply (1) by 4 and (2) by 3: 8u + 12v = 52 ... (3) 15u - 12v = -6 ... (4) Step 2: Add (3) and (4):…

Card 7Word Problems

The sum of two numbers is 25. The larger number is 3 more than twice the smaller number. Find both numbers.

Answer

Let x = smaller number, y = larger number Step 1: Form equations: x + y = 25 ... (1) y = 2x + 3 ... (2) Step 2: Substitute (2) into (1): x + (2x + 3) = 25 Step 3: Solve: 3x + 3 = 25 → 3x = 22 → x = …

Card 8Method Selection

When do you use elimination method instead of substitution method?

Answer

Use elimination method when: 1. Both equations have simple integer coefficients 2. Making coefficients equal is straightforward 3. Substitution leads to complex fractions Example: 2x + 3y = 7 and 3x …

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