What is a Unique Solution? | Simple Explanation for Class 12

S7-SA2-0039

What is a Unique Solution of Linear Equations?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

A unique solution for a system of linear equations means there is only one specific set of values for the variables that makes all the equations true at the same time. Imagine finding one perfect meeting point for two straight roads; that's a unique solution.

Simple Example

Quick Example

Suppose you buy 2 samosas and 1 chai for Rs 50, and your friend buys 1 samosa and 1 chai for Rs 35. If we want to find the exact price of one samosa and one chai, there will be only one correct price for each. That single pair of prices is the unique solution.

Worked Example

Step-by-Step

Let's find the unique solution for these equations:
Equation 1: x + y = 10
Equation 2: x - y = 2

Step 1: Add Equation 1 and Equation 2.
(x + y) + (x - y) = 10 + 2
---
Step 2: Simplify the equation.
x + y + x - y = 12
2x = 12
---
Step 3: Solve for x.
x = 12 / 2
x = 6
---
Step 4: Substitute the value of x (which is 6) into Equation 1.
6 + y = 10
---
Step 5: Solve for y.
y = 10 - 6
y = 4
---
Answer: The unique solution is x = 6 and y = 4.

Why It Matters

Understanding unique solutions helps engineers design stable bridges, scientists predict weather patterns accurately, and even helps AI systems make precise decisions. It's crucial in fields like FinTech for calculating loan interests and in Medicine for dosage calculations, ensuring everything works perfectly.

Common Mistakes

MISTAKE: Assuming there will always be a unique solution for any two equations. | CORRECTION: Not all systems of equations have a unique solution. Some might have no solution (parallel lines) or infinitely many solutions (same line).

MISTAKE: Making calculation errors when substituting values back into equations. | CORRECTION: Always double-check your arithmetic, especially when adding, subtracting, or multiplying negative numbers. Verify your solution by putting both x and y values into ALL original equations.

MISTAKE: Confusing the unique solution with just finding one variable. | CORRECTION: A unique solution means finding a specific value for EVERY variable in the system that satisfies ALL equations simultaneously, not just one.

Practice Questions

Try It Yourself

QUESTION: Find the unique solution for: x + 2y = 7 and x - 2y = 3. | ANSWER: x = 5, y = 1

QUESTION: The sum of two numbers is 15, and their difference is 3. Find the two numbers. | ANSWER: The numbers are 9 and 6.

QUESTION: If 3 pens and 2 notebooks cost Rs 100, and 1 pen and 1 notebook cost Rs 40. What is the unique cost of one pen and one notebook? | ANSWER: Cost of one pen = Rs 20, Cost of one notebook = Rs 20.

MCQ

Quick Quiz

Which of these describes a system of linear equations with a unique solution?

The lines are parallel and never intersect.

The lines overlap completely, forming the same line.

The lines intersect at exactly one point.

The equations have more variables than equations.

The Correct Answer Is:

A unique solution means there is only one specific point (x, y) that satisfies both equations. Graphically, this happens when two lines intersect at exactly one point. Parallel lines never intersect (no solution), and overlapping lines intersect at infinitely many points (infinite solutions).

Real World Connection

In the Real World

Imagine you're using a GPS app like Google Maps or MapMyIndia. When it calculates the shortest route from your home in Delhi to a specific shop in Chandni Chowk, it's solving a complex system of equations to find a unique, optimal path. This involves finding unique intersection points and distances to give you precise directions.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation that forms a straight line when graphed, like y = mx + c | SYSTEM OF EQUATIONS: A collection of two or more equations with the same set of variables | VARIABLE: A symbol (like x or y) representing an unknown value | INTERSECTION POINT: The single point where two lines cross each other on a graph | SOLUTION: The values of variables that make an equation or system of equations true.

What's Next

What to Learn Next

Great job understanding unique solutions! Next, you can explore what happens when a system of linear equations has 'no solution' or 'infinitely many solutions'. This will give you a complete picture of how linear equations behave and prepare you for more complex problems in higher classes.