Equations are mathematical statements showing equality between two expressions. They involve variables constants and operators. Solving equations means finding values of variables that satisfy the equality. Linear quadratic and simultaneous equations are common types. Understanding equations is essential for algebra geometry and real-life problem-solving as they form the foundation of mathematics and help in analyzing relationships between different quantities.
Class 10 Maths Chapter 3 focuses on Pair of Linear Equations in Two Variables. It covers graphical and algebraic methods like substitution elimination and cross-multiplication to solve equations. The chapter emphasizes consistency of equations and real-life applications. Designed for CBSE Exam 2024-25 it strengthens problem-solving skills and prepares students for advanced topics in mathematics by building a clear understanding of linear relationships and their solutions.
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Step 1: Equation Analysis
Equation ₁: 2x + 3y = 5
Equation ₂: 4x + 6y = 10
Step 2: Proportionality Check
– Coefficient of x in Equation ₁: 2
– Coefficient of x in Equation ₂: 4
– Ratio of x coefficients: 4 ÷ 2 = 2
– Coefficient of y in Equation ₁: 3
– Coefficient of y in Equation ₂: 6
– Ratio of y coefficients: 6 ÷ 3 = 2
Step 3: Constant Term Verification
– Equation ₁ constant: 5
– Equation ₂ constant: 10
– Ratio of constants: 10 ÷ 5 = 2
Step 4: Algebraic Manipulation
Divide Equation ₂ by 2:
2x + 3y = 5 (Identical to Equation ₁)
Step 5: Interpretation
– The equations represent EXACTLY THE SAME LINE
– This means the system has INFINITELY MANY SOLUTIONS
– Every point on this line satisfies both equations
Mathematical Insight:
When two linear equations represent identical lines,
they have an infinite number of solution points that
perfectly overlap each other.
Conclusion:
The system has INFINITELY MANY SOLUTIONS.
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