An equation is inconsistent when it has no solution. This happens if the lines represented by the equations are parallel and never intersect. Inconsistent equations have different slopes but the same variable coefficients. For example 2x + 3y = 5 and 4x + 6y = 12 are inconsistent as they cannot be satisfied simultaneously. Such systems are unsolvable.
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 explains consistent inconsistent and dependent systems. Practicing these concepts prepares students for CBSE Exam 2024-25 and strengthens problem-solving skills ensuring a clear understanding of real-life applications and mathematical foundations.
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Step 1: Definition of Inconsistent Equations
– Inconsistent equations are linear equations that have NO SOLUTION
– Graphically, this means the lines representing these equations NEVER intersect
Step 2: Graphical Interpretation
Inconsistent equations always result in PARALLEL LINES
– These lines have the same slope but different y-intercepts
– They run alongside each other, maintaining a constant distance
– No point exists where these lines cross
Step 3: Algebraic Characteristics
Example of Inconsistent Equations:
– Equation ₁: 2x + y = 4
– Equation ₂: 2x + y = 6
Observe:
– Same coefficient for x (2)
– Same coefficient for y (1)
– Different constant terms (4 and 6)
– This guarantees the lines will be parallel
Step 4: Geometric Visualization
– Imagine two identical lines shifted vertically
– They have the same “direction” but never touch
– No matter how far you extend them, they remain parallel
Conclusion:
For a pair of linear equations to be inconsistent, their graphs MUST be PARALLEL LINES.
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