What is Pair of Linear Equations in Two Variables Class 10: Definition & Concepts

Definition of Pair of Linear Equations in Two Variables

A pair of linear equations in two variables consists of two equations where each equation is linear and involves the same two variables, usually $x$ and $y$. The general form is:

$$ \begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases} $$

Here, $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are constants, and $x$, $y$ are variables. The solution to this pair is the set of values of $x$ and $y$ that satisfy both equations simultaneously.

In Class 10 NCERT, understanding this definition is foundational before learning solution methods.

Graphical Method to Solve Pair of Linear Equations

The graphical method involves plotting both linear equations on the Cartesian plane and finding their point(s) of intersection.

• Each equation represents a straight line.
• The solution corresponds to the point where both lines meet.

Steps:

1. Rewrite each equation in the form $y = mx + c$ or find two points for each line. 2. Plot both lines on graph paper. 3. Identify the intersection point.

Example:

Consider the pair:

$$ \begin{cases} x + y = 5 \\ 2x - y = 4 \end{cases} $$

• For $x + y = 5$, points are $(0,5)$ and $(5,0)$.
• For $2x - y = 4$, points are $(0,-4)$ and $(2,0)$.

Plotting these lines, their intersection gives the solution $(3,2)$.

This method helps visualise solutions but may be less precise for complex equations.

Algebraic Methods: Substitution and Elimination

Besides graphical solutions, Class 10 NCERT teaches two main algebraic methods:

1. Substitution Method

• Solve one equation for one variable.
• Substitute this value into the other equation.
• Solve the resulting single-variable equation.

Example:

$$ \begin{cases} x + y = 7 \\ 2x - y = 3 \end{cases} $$

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

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

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

2. Elimination Method

• Add or subtract equations to eliminate one variable.
• Solve for the remaining variable.
• Substitute back to find the other variable.

Example:

$$ \begin{cases} 3x + 2y = 16 \\ 5x - 2y = 4 \end{cases} $$

Add both:

$$8x = 20 \Rightarrow x = \frac{20}{8} = \frac{5}{2}$$

Substitute in first:

$$3 \times \frac{5}{2} + 2y = 16 \Rightarrow \frac{15}{2} + 2y = 16 \Rightarrow 2y = 16 - \frac{15}{2} = \frac{17}{2} \Rightarrow y = \frac{17}{4}$$

Both methods are essential for Class 10 exams.

Types of Solutions for Pair of Linear Equations

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

Key points:

• Unique solution means one ordered pair $(x,y)$ satisfies both.
• No solution means the system is inconsistent.
• Infinite solutions mean the two equations represent the same line.

Understanding these types helps solve and interpret problems correctly.

Real-Life Applications of Pair of Linear Equations

Pair of linear equations in two variables is not just theoretical; it applies in many real-life situations:

• Business: Calculating cost and revenue to find break-even points.
• Physics: Relating speed, distance, and time in motion problems.
• Economics: Supply and demand analysis using equations.
• Daily Life: Sharing expenses, mixing solutions with different concentrations.

Example:

If two shops sell pens and pencils at different prices, equations can help find quantities bought given total cost.

These applications make the chapter important for Class 10 students to understand practical maths.

Summary of Important Formulas and Concepts

Here is a quick reference table of key formulas and concepts:

Mastering these helps you solve questions efficiently in Class 10 exams.