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

Definition of Unique Solution in Linear Equations for Class 9

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, if you have two equations:

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

Solving these will give a single pair $(x, y)$ that works for both. This unique pair is called the unique solution.

In Class 9 NCERT mathematics, this concept helps students understand how two lines behave on a graph and how their intersection relates to solutions.

How to Identify a Unique Solution Using Graphs

Graphically, a unique solution appears when two lines intersect at exactly one point.

• Each linear equation represents a straight line on the coordinate plane.
• If the lines cross at one point, the coordinates of that point are the unique solution.

Example:

Plotting the equations:

$$ x + y = 4 \\ 2x - y = 1 $$

The lines intersect at a single point, say $(x_0, y_0)$, which is the unique solution.

If lines are parallel, no solution exists; if they coincide, infinite solutions exist.

Conditions for Unique Solution in Two Variables

For two linear equations:

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

The system has a unique solution if:

$$ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} $$

This means the coefficients of $x$ and $y$ are not proportional, so the lines have different slopes and intersect once.

Solving Linear Equations to Find Unique Solution: Worked Example

Let's solve the system:

$$ 3x + 2y = 16 \\ 2x - y = 3 $$

Step 1: Express $y$ from the second equation:

$$ y = 2x - 3 $$

Step 2: Substitute in the first equation:

$$ 3x + 2(2x - 3) = 16 \\ 3x + 4x - 6 = 16 \\ 7x = 22 \\ x = \frac{22}{7} $$

Step 3: Find $y$:

$$ y = 2 \times \frac{22}{7} - 3 = \frac{44}{7} - 3 = \frac{44}{7} - \frac{21}{7} = \frac{23}{7} $$

Solution:

$$ \left( \frac{22}{7}, \frac{23}{7} \right) $$

This is the unique solution satisfying both equations.

Difference Between Unique, No, and Infinite Solutions

Understanding the types of solutions helps clarify what unique solution means:

This table helps Class 9 students quickly identify the solution type by comparing coefficients.

Importance of Unique Solution in Class 9 NCERT Mathematics

The concept of unique solution is fundamental in the Class 9 NCERT chapter on linear equations in two variables because:

• It forms the basis for solving real-life problems involving two unknowns.
• Helps in understanding graphical representation of equations.
• Prepares students for higher classes where systems of equations become more complex.
• Enhances problem-solving skills by applying algebraic and graphical methods.

Students should practice various examples and exercises from the NCERT textbook to master this concept for exams.