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What is the Unique Solution of a System?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A unique solution for a system of equations means there is only one specific set of values for the variables that makes all equations in the system true. It's like finding the one exact meeting point for all the lines or conditions given.
Simple Example
Quick Example
Imagine you have two friends, Priya and Rahul, and you know two things: 1) Priya's age plus Rahul's age is 20, and 2) Priya's age is exactly double Rahul's age. There's only one possible age for Priya and one for Rahul that fits both these conditions. Priya must be 10 and Rahul must be 10 to satisfy the first condition. But for the second condition Priya must be double Rahul's age. So the only unique solution is Priya is 40 and Rahul is 20. Wait, that does not satisfy the first condition. Let's re-do the example. Priya's age plus Rahul's age is 30. Priya's age is double Rahul's age. The only unique solution is Priya is 20 and Rahul is 10.
Worked Example
Step-by-Step
Let's find the unique solution for this system of two linear equations:
Equation 1: x + y = 10
Equation 2: x - y = 2
Step 1: We want to eliminate one variable. Add Equation 1 and Equation 2.
(x + y) + (x - y) = 10 + 2
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Step 2: Simplify the equation.
2x = 12
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Step 3: Solve for x.
x = 12 / 2
x = 6
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Step 4: Substitute the value of x (which is 6) into either Equation 1 or Equation 2. Let's use Equation 1.
6 + y = 10
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Step 5: Solve for y.
y = 10 - 6
y = 4
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Step 6: Check if these values satisfy both original equations.
For Equation 1: 6 + 4 = 10 (True)
For Equation 2: 6 - 4 = 2 (True)
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Answer: The unique solution is x = 6 and y = 4.
Why It Matters
Understanding unique solutions is crucial in fields like AI/ML for optimizing algorithms, in Physics for calculating precise trajectories, and in Engineering for designing stable structures. Doctors use it to determine precise medicine dosages, and space scientists at ISRO use it for accurate satellite placement.
Common Mistakes
MISTAKE: Not checking if the solution satisfies ALL equations. Students often solve for one variable and stop, or only check one equation. | CORRECTION: Always substitute your found values back into EVERY original equation to ensure they all hold true. This confirms it's a valid solution for the entire system.
MISTAKE: Making calculation errors during elimination or substitution. Simple addition/subtraction mistakes can lead to a wrong final answer. | CORRECTION: Be very careful with your arithmetic. Double-check each step, especially when dealing with negative numbers or fractions.
MISTAKE: Confusing unique solutions with 'no solution' or 'infinitely many solutions'. Students might assume every system has a unique answer. | CORRECTION: Remember that a system can have no solution (parallel lines) or infinitely many solutions (overlapping lines). A unique solution means the lines intersect at exactly one point.
Practice Questions
Try It Yourself
QUESTION: Find the unique solution for: x + 2y = 7 and 3x - 2y = 5. | ANSWER: x = 3, y = 2
QUESTION: The cost of 2 samosas and 1 chai is Rs 40. The cost of 1 samosa and 2 chais is Rs 35. Find the unique cost of one samosa and one chai. | ANSWER: Samosa = Rs 15, Chai = Rs 10
QUESTION: Solve for a and b: 2a + 3b = 13 and 4a - b = 5. | ANSWER: a = 2, b = 3
MCQ
Quick Quiz
Which of the following describes a system of linear equations with a unique solution?
The lines are parallel and never intersect.
The lines overlap completely, being the same line.
The lines intersect at exactly one point.
The equations have no variables.
The Correct Answer Is:
C
A unique solution means there is exactly one set of values that satisfies all equations. Graphically, this happens when the lines representing the equations intersect at a single, distinct point.
Real World Connection
In the Real World
When you use a navigation app like Google Maps or Ola Cabs, the app solves a complex system of equations to find the 'unique' shortest or fastest route from your current location to your destination, considering traffic, road closures, and multiple paths.
Key Vocabulary
Key Terms
SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | VARIABLE: A letter (like x or y) representing an unknown value | SOLUTION: The value(s) of the variable(s) that make an equation true | ELIMINATION METHOD: A way to solve systems by adding or subtracting equations to remove a variable | SUBSTITUTION METHOD: A way to solve systems by solving one equation for a variable and plugging it into another equation
What's Next
What to Learn Next
Next, explore 'Systems with No Solution' and 'Systems with Infinitely Many Solutions'. Understanding these will complete your knowledge of all possible outcomes when solving systems of linear equations, preparing you for more complex problems.
