If the system of equations 2x + 3y = 5, 4x + ky = 10 has infinitely many solutions, then k =
A system of equations consists of two or more equations with the same variables. The solution satisfies all equations simultaneously. It can have unique solutions infinite solutions or no solution based on the relationship between the equations. Solving such systems helps in real-life applications like finding unknowns in science economics and engineering problems.
Class 10 Maths Chapter 3 deals with Pair of Linear Equations in Two Variables. It includes solving equations using substitution elimination and cross-multiplication methods. The chapter highlights graphical solutions and consistency conditions. These concepts are vital for real-world applications and enhance problem-solving skills. Preparing this chapter thoroughly is essential for the CBSE Exam 2024-25 as it strengthens algebraic understanding and supports advanced mathematical learning.
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Step 1: Understanding Infinitely Many Solutions
• In a system of linear equations, infinitely many solutions occur when the equations represent the same line.
• This means the equations are scalar multiples of each other.
Step 2: Mathematical Representation
Given equations:
1. 2x + 3y = 5 (Equation ₁)
2. 4x + ky = 10 (Equation ₂)
Step 3: Condition for Infinitely Many Solutions
For the lines to be the same, the coefficients must maintain a consistent ratio.
Coefficient Analysis:
• 1st equation: 2x + 3y = 5
– Coefficient of x: 2
– Coefficient of y: 3
• 2nd equation: 4x + ky = 10
– Coefficient of x: 4
– Coefficient of y: k
Step 4: Ratio Consistency
Coefficient of x ratio: 4 ÷ 2 = 2
Coefficient of y ratio: k ÷ 3 = 2
Therefore:
k = 3 * 2 = 6
Step 5: Verification
• When k = 6, the second equation becomes: 4x + 6y = 10
• Dividing by 2: 2x + 3y = 5 (exactly the same as the first equation)
Conclusion:
The value of k must be 6 to create infinitely many solutions.
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