What is a System of Linear Equations with No Solution?

S3-SA1-0282

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A system of linear equations with no solution is a set of two or more equations where you cannot find any values for the variables that satisfy all equations at the same time. Imagine trying to find a meeting point for two roads that are perfectly parallel – they will never meet, just like these equations have no common solution.

Simple Example

Quick Example

Imagine your elder brother says, 'I bought 2 samosas and 1 chai for Rs 50.' Then your sister says, 'I bought 4 samosas and 2 chai for Rs 110.' If the samosas and chai cost the same amount in both cases, this system would have no solution because the prices don't match up proportionally, making it impossible for both statements to be true.

Worked Example

Step-by-Step

Let's look at this system of 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.
---Step 2: Look at Equation 1. It says x plus y equals 5.
---Step 3: Now look at Equation 2. It says x plus y equals 7.
---Step 4: Can the same two numbers (x and y) add up to 5 AND also add up to 7 at the exact same time?
---Step 5: No, this is impossible! If x + y is 5, it cannot also be 7.
---Step 6: Therefore, there are no values for x and y that can satisfy both equations simultaneously.
---Answer: This system of linear equations has no solution.

Why It Matters

Understanding systems with no solution helps engineers design stable structures and computer scientists write efficient code. Data scientists use this concept when analyzing data to identify inconsistencies or impossible scenarios, ensuring their models are accurate. It's a fundamental idea in fields like AI/ML, cryptography, and even economics to understand limitations and constraints.

Common Mistakes

MISTAKE: Assuming there must always be a solution and trying to force one | CORRECTION: Recognize that sometimes equations are inconsistent and no solution exists. Look for contradictions.

MISTAKE: Only checking one equation and not the other | CORRECTION: A solution must satisfy ALL equations in the system simultaneously. Check every equation.

MISTAKE: Making calculation errors that hide the 'no solution' case | CORRECTION: Double-check your arithmetic carefully. Simple errors can lead you to think there's a solution when there isn't, or vice-versa.

Practice Questions

Try It Yourself

QUESTION: Does the system x + 2y = 10 and x + 2y = 12 have a solution? | ANSWER: No, it has no solution.

QUESTION: What type of solution does the system 3x - y = 5 and 6x - 2y = 8 have? | ANSWER: No solution. (Hint: Multiply the first equation by 2. You get 6x - 2y = 10, which contradicts 6x - 2y = 8).

QUESTION: Imagine two friends, Amit and Priya. Amit says, 'I have 5 more candies than Priya.' Priya says, 'I have 7 more candies than Amit.' Can both statements be true at the same time? Why or why not? | ANSWER: No, both statements cannot be true. If Amit has 5 more than Priya, Priya cannot also have 7 more than Amit. This is a system with no solution.

MCQ

Quick Quiz

Which of the following systems of equations has no solution?

x + y = 5 and x - y = 1

2x + 3y = 7 and 2x + 3y = 9

x + y = 5 and 2x + 2y = 10

x = 3 and y = 2

The Correct Answer Is:

In option B, 2x + 3y cannot be equal to both 7 and 9 at the same time, making it a system with no solution. Options A, C, and D all have solutions.

Real World Connection

In the Real World

Imagine you're a traffic police officer using a GPS system to manage traffic flow in a busy Indian city. If two traffic rules, when converted into equations, show that a vehicle must be at two different places at the exact same time, the system would have no solution. This indicates a contradiction or an impossible scenario, which helps in identifying faulty data or planning better traffic routes.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | LINEAR EQUATION: An equation whose graph is a straight line | SOLUTION: A value or set of values for the variables that makes the equation(s) true | INCONSISTENT SYSTEM: A system of equations that has no solution

What's Next

What to Learn Next

Great job understanding when equations don't have a solution! Next, you can learn about 'Systems of Linear Equations with One Solution' and 'Systems of Linear Equations with Infinite Solutions.' This will complete your understanding of all possible outcomes when solving systems of equations.