Pair of Linear Equations in Two Variables – Detailed Explanation for Class 10
The chapter Pair of Linear Equations in Two Variables is one of the most important topics in Class 10 Mathematics. This chapter introduces students to the idea of solving two equations simultaneously to find the values of two unknown variables. These variables are usually represented by x and y. The concept not only helps in scoring good marks in board examinations but also develops logical thinking and problem-solving skills that are useful in real life.
What Is a Linear Equation in Two Variables?
A linear equation in two variables is an equation that can be written in the form:
where a, b, and c are real numbers and a and b are not both zero.
Here, x and y are the variables whose values we want to find.
When two such linear equations are taken together, they form a pair of linear equations in two variables. For example:
These two equations together help us find a unique solution for x and y.
Graphical Representation of Linear Equations
Each linear equation in two variables represents a straight line when plotted on a graph. Therefore, a pair of linear equations represents two straight lines on the same coordinate plane.
Based on how these two lines behave, the system of equations can have different types of solutions:
Intersecting Lines
When two lines intersect at exactly one point, the pair of linear equations has one unique solution.
Parallel Lines
When two lines never intersect, the pair of linear equations has no solution. Such equations are called inconsistent.
Coincident Lines
When both lines overlap completely, there are infinitely many solutions. These equations are called dependent.
Algebraic Methods of Solving Pair of Linear Equations
Apart from the graphical method, Class 10 syllabus focuses on algebraic methods for accurate and quick solutions.
In the substitution method, one variable is expressed in terms of the other from one equation and then substituted into the second equation.
Choose one equation and express one variable in terms of the other.
Substitute this value in the second equation.
Solve the resulting equation.
Substitute the value back to find the second variable.
This method is useful when one variable has a coefficient of 1 or −1.
The elimination method involves eliminating one variable by adding or subtracting the two equations.
Make the coefficients of one variable equal in both equations.
Add or subtract the equations to eliminate that variable.
Solve for the remaining variable.
Substitute the value to find the second variable.
This is the most commonly used method in board examinations due to its simplicity and speed.
3. Cross-Multiplication Method
The cross-multiplication method is a direct formula-based approach.
If the equations are:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
y = (a₂c₁ − a₁c₂) / (a₁b₂ − a₂b₁)
This method is quick but requires careful calculation.
Conditions for Consistency of Pair of Linear Equations
To check whether a pair of linear equations has a solution or not, we compare the ratios:
a₁/a₂ ≠ b₁/b₂ → One unique solution
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution
a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinitely many solutions
Understanding these conditions is very important for solving theoretical questions in exams.
Real-Life Applications of Pair of Linear Equations
Pair of linear equations are widely used in daily life situations such as:
Calculating the cost of two items
Finding speed, distance, and time
Solving problems related to age
Business profit and loss calculations
Comparing quantities in mixtures and ratios
These applications show how mathematics connects with real-world problems.
Importance for Class 10 Board Exams
This chapter carries significant weightage in board exams. Questions are asked in the form of:
Algebraic solution problems
Regular practice of NCERT questions and examples is essential to score well.
Practice all three solving methods
Learn graphical concepts clearly
Write steps neatly in exams
Double-check calculations
The chapter Pair of Linear Equations in Two Variables is a foundation topic in Class 10 Mathematics. It strengthens algebraic skills and helps students understand the relationship between equations and graphs. With consistent practice and clear understanding, students can easily master this chapter and perform confidently in exams.
