What Is Pair of Linear Equations in Two Variables? Class 10 NCERT Guide

Definition and Basic Form of Pair of Linear Equations in Two Variables

A pair of linear equations in two variables refers to two equations that involve the same two variables, usually $x$ and $y$. Each equation is linear, meaning the variables are to the power of one and appear only in the first degree. The general form of each equation is:

$$ a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 $$

Here, $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are constants, and $x$, $y$ are variables. The goal is to find values of $x$ and $y$ that satisfy both equations simultaneously.

In Class 10 NCERT Mathematics, understanding this definition is crucial as it forms the foundation for solving systems of equations and applying them to real-world problems.

Graphical Interpretation: How to Visualize Pair of Linear Equations

Graphically, each linear equation in two variables represents a straight line on the Cartesian plane. When you plot both lines:

• If the lines intersect at a single point, that point $(x, y)$ is the unique solution.
• If the lines coincide (are the same), there are infinitely many solutions.
• If the lines are parallel and distinct, there is no solution.

Example:

Consider the pair:

$$ 2x + 3y = 6 \\ 4x - y = 5 $$

Plotting these lines helps us find their intersection point, which gives the solution to the pair of equations. This visual method aids in understanding the nature of solutions.

Methods to Solve Pair of Linear Equations in Two Variables

Class 10 students learn several methods to solve these pairs:

• Substitution Method: Solve one equation for one variable, then substitute into the other.
• Elimination Method: Add or subtract equations to eliminate one variable.
• Graphical Method: Plot both lines and find the intersection.

Worked Example (Substitution):

Solve:

$$ x + y = 7 \\ 2x - y = 3 $$

From first, $y = 7 - x$. Substitute in second:

$$ 2x - (7 - x) = 3 \\ 2x - 7 + x = 3 \\ 3x = 10 \\ x = \frac{10}{3} $$

Then, $y = 7 - \frac{10}{3} = \frac{21 - 10}{3} = \frac{11}{3}$.

Solution: $\left( \frac{10}{3}, \frac{11}{3} \right)$.

Types of Solutions and Their Conditions

A pair of linear equations in two variables can have three types of solutions:

Understanding these conditions helps students quickly identify the nature of solutions without graphing.

Real-Life Applications of Pair of Linear Equations in Two Variables

Pair of linear equations are not just theoretical; they model many real-life situations such as:

• Calculating costs and profits
• Mixing solutions with different concentrations
• Finding speed, distance, and time in travel problems
• Distributing items in given ratios

For example, if a shopkeeper sells pens and pencils and wants to find how many of each were sold given total items and total cost, a pair of linear equations can be formed and solved.

Class 10 NCERT problems often include such applications to develop problem-solving skills.

Important Formulas and Tips for Class 10 NCERT Students

Key formulas and tips:

• General form: $a_1x + b_1y = c_1$, $a_2x + b_2y = c_2$
• Use substitution or elimination methods for accuracy
• Check solutions by substituting back into both equations
• For graphical method, plot at least two points per line
• Understand conditions for types of solutions

Tip: Always simplify equations before solving to avoid errors.

Formula for elimination: Multiply equations to make coefficients equal, then add or subtract:

$$ \text{If } a_1 = a_2, \quad (a_1x + b_1y) - (a_2x + b_2y) = c_1 - c_2 $$