What is System of Equations with Unique Solution? | Simple Explanation for Class 12

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What is the System of Equations with Unique Solution?

Grade Level:

Class 12

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

Definition

What is it?

A system of equations with a unique solution means there's only one specific set of values for all the unknown variables that makes all the equations true at the same time. Think of it like finding one exact meeting point for several different roads.

Simple Example

Quick Example

Imagine you buy two samosas and one chai for Rs 50. Your friend buys one samosa and two chais for Rs 40. If there's a unique solution, it means there's only one possible price for a samosa and one possible price for a chai that satisfies both these situations.

Worked Example

Step-by-Step

Let's find the unique solution for this system:
Equation 1: x + y = 7
Equation 2: x - y = 3

---Step 1: Add Equation 1 and Equation 2.
(x + y) + (x - y) = 7 + 3
2x = 10

---Step 2: Solve for x.
x = 10 / 2
x = 5

---Step 3: Substitute the value of x (which is 5) into Equation 1.
5 + y = 7

---Step 4: Solve for y.
y = 7 - 5
y = 2

---Step 5: Check your solution by substituting x=5 and y=2 into Equation 2.
5 - 2 = 3 (This is true!)

---Answer: The unique solution is x = 5 and y = 2.

Why It Matters

Understanding unique solutions helps in fields like AI/ML to train models, in Physics to predict object movements, and in Finance to calculate investments. Engineers use this to design structures, and data scientists use it to make sense of complex data, creating many exciting career paths.

Common Mistakes

MISTAKE: Not checking the solution in all original equations. | CORRECTION: Always substitute your found values back into ALL the original equations to ensure they are all satisfied.

MISTAKE: Making calculation errors when adding or subtracting equations. | CORRECTION: Double-check your arithmetic, especially when dealing with negative signs. Work slowly and carefully.

MISTAKE: Confusing unique solution with no solution or infinite solutions. | CORRECTION: A unique solution means one specific (x,y) pair. No solution means no such pair exists. Infinite solutions mean many pairs work.

Practice Questions

Try It Yourself

QUESTION: Find the unique solution for: x + 2y = 10 and x - y = 1. | ANSWER: x = 4, y = 3

QUESTION: A shop sells pens and notebooks. 3 pens and 2 notebooks cost Rs 70. 1 pen and 1 notebook cost Rs 25. Find the cost of one pen and one notebook. | ANSWER: Pen = Rs 20, Notebook = Rs 5

QUESTION: Solve for a and b: 2a + 3b = 13 and 4a - b = 5. | ANSWER: a = 2, b = 3

MCQ

Quick Quiz

Which statement best describes a system of equations with a unique solution?

There are no values that satisfy all equations.

There is exactly one set of values that satisfies all equations.

There are many sets of values that satisfy all equations.

The equations are always parallel lines.

The Correct Answer Is:

A unique solution means there is only one specific set of values for the variables that makes all equations true. Options A and C describe 'no solution' and 'infinite solutions' respectively.

Real World Connection

In the Real World

When a satellite like those used by ISRO for navigation (NavIC) needs to pinpoint its exact location, it uses signals from multiple ground stations. Each signal provides an equation, and solving this system of equations with a unique solution gives the satellite's precise coordinates in space.

Key Vocabulary

Key Terms

VARIABLE: A letter representing an unknown value, like 'x' or 'y' | EQUATION: A mathematical statement showing two expressions are equal | SYSTEM OF EQUATIONS: A collection of two or more equations with the same variables | SOLUTION: The set of values for variables that makes an equation or system true | UNIQUE: Being the only one of its kind.

What's Next

What to Learn Next

Now that you understand unique solutions, you can explore systems with 'no solution' and 'infinite solutions'. This will help you see all possibilities when solving equations and prepare you for advanced topics like matrices and determinants.