If 3x + 4y = 10 and 6x + ky = 20 have a unique solution, then the value of k must not be:
A unique solution in mathematics refers to a situation where a system of equations or a problem has exactly one solution. For linear equations this happens when the lines intersect at a single point. It ensures clear and specific results without ambiguity. Understanding unique solutions is crucial for solving real-life problems and advanced mathematical concepts effectively.
Class 10 Maths Chapter 3 focuses on Pair of Linear Equations in Two Variables. It covers methods like substitution elimination and graphical representation to solve equations. The chapter emphasizes consistency and inconsistency of solutions. Students will learn to apply these concepts to real-life problems. Preparing for CBSE Exam 2024-25 this chapter strengthens analytical skills and problem-solving abilities ensuring a solid foundation for advanced mathematics.
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Step 1: Understanding Unique Solution Conditions
• A system of two linear equations has a unique solution when the equations are linearly independent.
• Linear independence means the lines represented by these equations are not parallel.
Step 2: Mathematical Representation
Given equations:
1. 3x + 4y = 10 (Equation ₁)
2. 6x + ky = 20 (Equation ₂)
Step 3: Condition for Unique Solution
For a unique solution, the coefficient matrix must have a non-zero determinant.
Coefficient matrix = [³⁄₁ ⁴⁄₁]
[⁶⁄₁ ᵏ⁄₁]
Step 4: Determinant Calculation
det = (3 * k) – (4 * 6)
= 3k – 24
Step 5: Uniqueness Condition
For unique solution, det ≠ 0
3k – 24 ≠ 0
3k ≠ 24
k ≠ 8
Step 6: Identifying Impossible k
The condition k = 8 makes the lines parallel, preventing a unique solution.
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