What is No Solution for a System?

S6-SA1-0228

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

When we have a 'system' of equations (two or more equations together), a 'no solution' situation means there are no values for the variables (like x and y) that can satisfy all the equations at the same time. Think of it as trying to find a common meeting point for two roads that are always parallel and never cross.

Simple Example

Quick Example

Imagine you have two friends, Priya and Rohan. Priya says, 'I am thinking of a number, and if you add 5 to it, you get 10.' Rohan says, 'I am thinking of the SAME number, and if you add 5 to it, you get 12.' It's impossible for the same number to give two different results when you add 5 to it. So, there is no solution to what number they are thinking of.

Worked Example

Step-by-Step

Let's consider a system of two linear equations:
Equation 1: x + y = 5
Equation 2: x + y = 7

Step 1: We want to find values for x and y that make both equations true. Let's try to eliminate one variable.
---Step 2: Subtract Equation 1 from Equation 2.
(x + y) - (x + y) = 7 - 5
---Step 3: Simplify both sides of the equation.
x + y - x - y = 2
---Step 4: The x terms cancel out, and the y terms cancel out.
0 = 2
---Step 5: We are left with the statement 0 = 2, which is false. This means there are no values of x and y that can satisfy both equations simultaneously.
Answer: The system has no solution.

Why It Matters

Understanding 'no solution' is crucial in fields like AI/ML for optimizing algorithms, in Physics for analyzing forces that can't balance, and in Engineering for designing stable structures. Engineers use this to avoid impossible designs, while data scientists might realize their models have conflicting requirements, helping them refine their approach.

Common Mistakes

MISTAKE: Assuming that if you can't find a solution quickly, it means there's no solution. | CORRECTION: Always try algebraic methods (substitution or elimination) to simplify the system. Only if you arrive at a false statement (like 0 = 5) can you conclude there is no solution.

MISTAKE: Confusing 'no solution' with 'infinite solutions'. | CORRECTION: 'No solution' means a false statement (e.g., 0 = 3). 'Infinite solutions' means you get a true statement where variables cancel out (e.g., 0 = 0), indicating the equations are essentially the same.

MISTAKE: Thinking that 'no solution' means the variables are zero. | CORRECTION: 'No solution' means there are NO numbers (zero or otherwise) that work for ALL equations. It doesn't mean the variables themselves are zero.

Practice Questions

Try It Yourself

QUESTION: Does the following system have a solution?
Equation 1: 2x + 3y = 6
Equation 2: 2x + 3y = 10
| ANSWER: No solution

QUESTION: Find if a solution exists for the system:
Equation 1: x - 2y = 4
Equation 2: 3x - 6y = 15
| ANSWER: No solution

QUESTION: A vendor sells samosas (x) and chai (y). On Monday, he sold 5 samosas and 3 chais for Rs 100. On Tuesday, he sold 10 samosas and 6 chais for Rs 250. Is this pricing consistent, or is there no solution for the price of each item?
| ANSWER: No solution (The equations would be 5x + 3y = 100 and 10x + 6y = 250. If you multiply the first equation by 2, you get 10x + 6y = 200, which contradicts the second equation.)

MCQ

Quick Quiz

Which of the following graphical representations shows a system with 'no solution'?

Two lines intersecting at one point

Two lines that are parallel and never meet

Two lines that are exactly on top of each other

A single curved line

The Correct Answer Is:

When lines are parallel and never meet, there is no common point (x, y) that satisfies both equations, meaning no solution. Intersecting lines have one solution, and overlapping lines have infinite solutions.

Real World Connection

In the Real World

Imagine traffic control systems in a smart city. If two routes are planned such that they are parallel but are supposed to lead to the same destination via different intermediate points with conflicting travel times, the system might identify this as a 'no solution' scenario. This helps city planners redesign routes to avoid congestion and ensure smooth flow, much like how ISRO scientists calculate trajectories to avoid impossible orbital paths.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | PARALLEL LINES: Lines that are always the same distance apart and never intersect | INCONSISTENT SYSTEM: Another name for a system with no solution | LINEAR EQUATION: An equation where the highest power of the variable is 1 (forms a straight line when graphed)

What's Next

What to Learn Next

Now that you understand 'no solution', next you should explore 'systems with one solution' and 'systems with infinite solutions'. This will complete your understanding of all possible outcomes when solving systems of linear equations, which is super helpful for more complex math and science problems!