What is System of Equations with No Solution? | Simple Explanation for Class 12

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What is the System of Equations with No 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 no solution means there is no set of values for the variables that can make all the equations true at the same time. When you try to solve such a system, you will end up with a false mathematical statement, like '0 = 5'.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya. Rahul says, 'I have 5 more cricket stickers than Priya.' Priya says, 'I have 2 more cricket stickers than Rahul.' If you try to find out how many stickers each has, you'll find it's impossible. Their statements contradict each other, meaning there's no solution.

Worked Example

Step-by-Step

Let's solve this system of equations:
Equation 1: x + y = 5
Equation 2: x + y = 8

1. We want to find values for 'x' and 'y' that satisfy both equations.
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2. Let's try to eliminate one variable. Subtract Equation 1 from Equation 2.
(x + y) - (x + y) = 8 - 5
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3. Simplify both sides of the equation.
x + y - x - y = 3
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4. Notice that 'x' and '-x' cancel out, and 'y' and '-y' cancel out.
0 = 3
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5. This statement '0 = 3' is false. It means there are no values of x and y that can make both original equations true simultaneously.

Answer: The system has no solution.

Why It Matters

Understanding systems with no solution is crucial in many fields! In AI/ML, it helps in identifying inconsistent data sets. In engineering, it can signal impossible design conditions for bridges or circuits. Engineers and data scientists use this concept to avoid building things that won't work or making incorrect predictions.

Common Mistakes

MISTAKE: Assuming there must be a solution and making calculation errors to force one. | CORRECTION: If you reach a false statement like '0 = 7', accept that the system has no solution and don't try to 'fix' it.

MISTAKE: Not checking if equations are equivalent or contradictory after simplification. | CORRECTION: Always simplify equations first and then compare them. If the left sides are identical but the right sides are different, it's a no-solution case.

MISTAKE: Confusing 'no solution' with 'infinite solutions'. | CORRECTION: 'No solution' means a false statement (e.g., 0=5). 'Infinite solutions' means the equations are identical or multiples of each other, leading to a true statement like '0=0'.

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: Determine if the following system has a solution: 3x - y = 7 and 6x - 2y = 10. | ANSWER: No, it has no solution. (Multiply the first equation by 2: 6x - 2y = 14. This contradicts the second equation.)

QUESTION: For what value of 'k' will the system 2x + 3y = 5 and 2x + ky = 7 have no solution? | ANSWER: k = 3. (If k=3, the left sides are identical but the right sides are different, leading to no solution.)

MCQ

Quick Quiz

Which of the following statements is true for a system of equations with no solution?

There is exactly one unique solution.

There are infinitely many solutions.

The equations contradict each other, leading to a false statement.

The variables cancel out, resulting in 0 = 0.

The Correct Answer Is:

A system with no solution means the equations are inconsistent and contradict each other, leading to a false mathematical statement like 0 = 5. Options A and B describe systems with solutions, while D describes a system with infinite solutions.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. If their routing software tries to create a path that requires a delivery truck to be in two different cities at the exact same time, the system will flag this as 'no solution' for that route. This helps them avoid impossible schedules and manage resources efficiently.

Key Vocabulary

Key Terms

SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | NO SOLUTION: No values for variables satisfy all equations simultaneously. | CONTRADICTION: A statement that is always false, like 0 = 5. | INCONSISTENT SYSTEM: Another name for a system of equations with no solution.

What's Next

What to Learn Next

Next, you should explore 'Systems of Equations with Infinite Solutions'. This will help you understand the full range of possibilities when solving multiple equations and build a strong foundation for advanced algebra and calculus.