Class 10 / Maths
Class 10 Maths Pair of Linear Equations Notes
Notes on solving pairs of linear equations by substitution, elimination, and cross multiplication.
Topic introduction
A pair of linear equations is useful for solving two unknown quantities together. The chapter tests algebraic methods, graphical interpretation, consistency of lines, and word-problem formation.
Key points
- A pair of linear equations in two variables represents two straight lines on a graph.
- Intersecting lines have one solution, parallel lines have no solution, and coincident lines have infinitely many solutions.
- Elimination is useful when coefficients can be made equal or opposite easily.
- Substitution is useful when one equation can quickly give x in terms of y or y in terms of x.
Definitions
Linear equation in two variables: An equation of the form ax + by + c = 0, where x and y are variables and a and b are not both zero.
Consistent pair: A pair of equations that has at least one solution.
Inconsistent pair: A pair of equations that has no solution.
Unique solution: A single ordered pair that satisfies both equations.
Formulas
- For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
- Unique solution: a1/a2 is not equal to b1/b2
- No solution: a1/a2 = b1/b2 is not equal to c1/c2
- Infinitely many solutions: a1/a2 = b1/b2 = c1/c2
Examples
- x + y = 9 and x - y = 3 gives x = 6 and y = 3.
- 2x + 3y = 5 and 4x + 6y = 10 represent coincident lines, so there are infinitely many solutions.
- A word problem about two prices usually starts by assigning x and y to the two unknown costs.
Graphical meaning
Each linear equation represents a line. The solution of the pair is the point where the two lines meet.
If the lines are parallel, there is no common point. If both equations represent the same line, there are infinitely many common points.
Algebraic methods
In substitution, express one variable using the other and substitute in the second equation.
In elimination, multiply equations if needed so that one variable cancels when equations are added or subtracted.
In cross multiplication, arrange equations in standard form and use the formula carefully.
Common mistakes
- Solving correctly but writing only x or only y instead of the ordered pair.
- Forgetting to multiply the whole equation while using elimination.
- Mixing up no solution and infinitely many solutions in ratio questions.
- Forming word-problem equations without clearly defining variables.
- Changing signs incorrectly when moving terms across the equal sign.
Quick practice
- Solve 2x + y = 7 and x - y = 2.
- Solve 3x + 2y = 11 and 2x - y = 3.
- Check whether x + 2y = 5 and 2x + 4y = 10 have infinitely many solutions.
- Form two equations for: The sum of two numbers is 35 and their difference is 7.
Related practice
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