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

Definition: What Is Pair of Linear Equations in Two Variables?

A pair of linear equations in two variables is a set of two linear equations, each involving the same two variables, usually denoted as $x$ and $y$. Each equation can be written in the form:

$$ax + by = c$$

where $a$, $b$, and $c$ are constants and $a$ and $b$ are not both zero. The pair together looks like:

$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

The goal is to find values of $x$ and $y$ that satisfy both equations simultaneously. This concept is fundamental in Class 10 NCERT Mathematics and is widely used in solving practical problems involving two unknowns.

Graphical Interpretation of Pair of Linear Equations

Each linear equation in two variables represents a straight line on the Cartesian plane. When you have a pair of such equations, their graphs can interact in three ways:

• Intersecting lines: The lines cross at exactly one point. This point $(x, y)$ is the unique solution to the pair.
• Parallel lines: The lines never meet, indicating no solution (inconsistent system).
• Coincident lines: The lines lie on top of each other, meaning infinitely many solutions.

Understanding this helps students visualize solutions and verify algebraic answers.

Methods to Solve Pair of Linear Equations in Two Variables

Class 10 NCERT Mathematics teaches three main methods to solve pairs of linear equations:

1. Graphical Method: Plot both equations on graph paper and find the intersection point.

2. Substitution Method: Solve one equation for one variable, substitute into the other, and solve.

3. Elimination Method: Add or subtract equations to eliminate one variable, then solve for the other.

Example: Solve using Substitution

Given:

$$\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$$

From first equation, $y = 5 - x$. Substitute into second:

$$2x - (5 - x) = 1 \Rightarrow 2x - 5 + x = 1 \Rightarrow 3x = 6 \Rightarrow x = 2$$

Then, $y = 5 - 2 = 3$. Solution: $(2, 3)$.

Practicing these methods ensures strong problem-solving skills.

Applications of Pair of Linear Equations in Real Life

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

• Business: Calculating cost and profit with two products.
• Travel: Finding speed and time relationships.
• Mixtures: Combining solutions or ingredients in certain ratios.
• Geometry: Determining points of intersection.

For example, if a shop sells pens and pencils, and you know the total number sold and total money earned, you can form two linear equations to find individual quantities.

This practical relevance makes the chapter important for Class 10 students preparing for CBSE exams.

Important Formulas and Tips for Class 10 Students

Here are key formulas and tips to remember:

• General form: $ax + by = c$
• Condition for unique solution: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
• Condition for infinite solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Condition for no solution: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

Tips:

• Always check coefficients ratios before solving.
• Draw neat graphs for visual understanding.
• Practice NCERT examples and exercises thoroughly.
• Use substitution or elimination when graphing is difficult.

These tips help students score well in exams and build a strong foundation.