What Is Unique Solution in Linear Equations in Two Variables Class 9 Explained

Definition of Unique Solution in Linear Equations

A unique solution in linear equations in two variables means there is exactly one ordered pair $(x, y)$ that satisfies both equations simultaneously. For example, consider two linear equations:

$$ \begin{cases} 2x + 3y = 6 \\ 4x - y = 5 \end{cases} $$

If there is only one pair of values $(x, y)$ that makes both equations true, then the system has a unique solution. This contrasts with systems having no solution or infinitely many solutions.

How to Identify a Unique Solution Graphically

Graphically, each linear equation in two variables represents a straight line on the coordinate plane. The system of two linear equations will have:

• Unique solution if the two lines intersect at exactly one point.
• No solution if the lines are parallel and never meet.
• Infinitely many solutions if the two lines coincide (are the same line).

For example, the lines represented by:

$$ \begin{cases} x + y = 4 \\ 2x - y = 1 \end{cases} $$

intersect at a single point, so there is a unique solution.

Algebraic Methods to Find Unique Solutions

To find the unique solution algebraically, you can use:

• Substitution Method: Solve one equation for one variable and substitute into the other.
• Elimination Method: Add or subtract equations to eliminate one variable.

Example (Substitution):

Solve:

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

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

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

Then, $y = 5 - 2 = 3$. Unique solution is $(2, 3)$.

Conditions for Unique, No, and Infinite Solutions

The nature of solutions depends on the coefficients of the equations:

Here, $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ are two linear equations.

Importance of Unique Solution in Class 9 NCERT Maths

Understanding what is unique solution in linear equations in two variables is crucial for Class 9 students because:

• It forms the foundation for solving real-life problems involving two variables.
• Helps in mastering algebraic techniques like substitution and elimination.
• Builds problem-solving skills needed for higher classes and competitive exams.
• Is a key topic in the NCERT textbook and CBSE exams.

Regular practice with NCERT exercises improves confidence and accuracy.

Worked Example: Finding Unique Solution Step-by-Step

Example: Solve the system:

$$ \begin{cases} 3x + 2y = 16 \\ 5x - y = 9 \end{cases} $$

Step 1: From second equation, express $y$:

$$y = 5x - 9$$

Step 2: Substitute in first equation:

$$3x + 2(5x - 9) = 16$$

$$3x + 10x - 18 = 16$$

$$13x = 34$$

$$x = \frac{34}{13} = 2.615$$

Step 3: Find $y$:

$$y = 5 \times 2.615 - 9 = 13.075 - 9 = 4.075$$

Solution: The unique solution is approximately $(2.615, 4.075)$.