Pair of Linear Equations in Two Variables Class 10 Maths — Revision Notes
This chapter focuses on systems of two linear equations with two variables. Students learn graphical and algebraic methods to solve them and understand con
TL;DR: This chapter focuses on systems of two linear equations with two variables. Students learn graphical and algebraic methods to solve them and understan…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
This chapter focuses on systems of two linear equations with two variables. Students learn graphical and algebraic methods to solve them and understand con
Linear Equation Form
- General form: ax + by + c = 0 where a, b, c are real and a, b not both zero
- Pair of linear equations: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
- Graph of linear equation is a straight line
- Solution is ordered pair (x, y) satisfying both equations
Graphical Method
- Plot both lines on coordinate plane
- If lines intersect at one point: unique solution (consistent)
- If lines coincide: infinite solutions (dependent consistent)
- If lines parallel: no solution (inconsistent)
Algebraic Methods
- Substitution method: solve one equation for variable, substitute in other
- Elimination method: multiply equations to make coefficient equal, then subtract
- Cross-multiplication method: for a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
- x = (b1c2 - b2c1)/(a1b2 - a2b1), y = (c1a2 - c2a1)/(a1b2 - a2b1)
Consistency and Ratios
- Consistent system has at least one solution
- Inconsistent system has no solution
- For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
- If a1/a2 = b1/b2 = c1/c2: infinite solutions
- If a1/a2 = b1/b2 ≠ c1/c2: no solution
- If a1/a2 ≠ b1/b2: unique solution
Key Terms
- Linear Equation: Equation of form ax + by + c = 0 representing a straight line
- Consistent System: System of equations having at least one solution
- Inconsistent System: System of equations having no solution
Frequently Asked Questions
How do you check if a pair of linear equations has unique solution?
Calculate the ratio a1/a2. If a1/a2 ≠ b1/b2, then the system has unique solution. Alternatively, find determinant: if a1b2 - a2b1 ≠ 0, unique solution exists.
What does it mean if lines are parallel?
If two lines are parallel, they never meet. So the system has no solution and is inconsistent. This occurs when a1/a2 = b1/b2 but ≠ c1/c2.
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