What is a Consistent System of Linear Equations?

S7-SA2-0037

Grade Level:

Class 12

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

Definition

What is it?

A consistent system of linear equations is a set of equations that has at least one solution. This means there are values for the variables that satisfy all equations simultaneously. Simply put, a consistent system always has an answer.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs 50. Your friend buys 4 samosas and 2 chai for Rs 100. This is a consistent system because we can find the exact price of one samosa and one chai that works for both situations. It has a solution!

Worked Example

Step-by-Step

Let's check if the system x + y = 5 and 2x + 2y = 10 is consistent.

STEP 1: Write down the equations: Equation (1): x + y = 5 | Equation (2): 2x + 2y = 10
---STEP 2: Try to simplify Equation (2). Divide both sides by 2: (2x + 2y) / 2 = 10 / 2, which gives x + y = 5.
---STEP 3: Notice that after simplification, Equation (2) is identical to Equation (1).
---STEP 4: This means both equations represent the same line. Any point on this line is a solution to both equations.
---STEP 5: For example, if x=2, then y=3 (since 2+3=5). This (2,3) is a solution for both.
---STEP 6: Since there are infinitely many solutions (all points on the line), the system has at least one solution.

ANSWER: The system is consistent.

Why It Matters

Understanding consistent systems helps engineers design stable structures and AI models learn patterns. In finance, it helps predict stock prices, and in space technology, it's used to calculate rocket trajectories, making careers in these fields exciting and impactful.

Common Mistakes

MISTAKE: Thinking that 'consistent' means only one unique solution. | CORRECTION: A consistent system can have one unique solution OR infinitely many solutions. The key is 'at least one solution'.

MISTAKE: Assuming a system is inconsistent if the equations look different. | CORRECTION: Always try to simplify equations or solve them before concluding consistency. Sometimes, different-looking equations are actually the same or parallel.

MISTAKE: Confusing the terms 'consistent' and 'independent'. | CORRECTION: 'Consistent' means it has solutions. 'Independent' refers to whether the equations provide unique information. A consistent system can have dependent equations (like two identical lines).

Practice Questions

Try It Yourself

QUESTION: Is the system x + y = 7 and x - y = 3 consistent? | ANSWER: Yes, it is consistent (x=5, y=2 is a solution).

QUESTION: Determine if the system 3x + 2y = 6 and 6x + 4y = 12 is consistent. | ANSWER: Yes, it is consistent (infinitely many solutions, as the second equation is twice the first).

QUESTION: Consider the system: x + y = 4, x - y = 2, and 2x + y = 7. Is this system consistent? | ANSWER: Yes, it is consistent (x=3, y=1 is a solution for all three equations).

MCQ

Quick Quiz

Which of the following describes a consistent system of linear equations?

It has no solution.

It has exactly one solution.

It has at least one solution (one or infinitely many).

It has infinitely many solutions only.

The Correct Answer Is:

A consistent system means there's always an answer. This answer can be unique (exactly one solution) or there can be countless answers (infinitely many solutions). So, 'at least one solution' covers both cases.

Real World Connection

In the Real World

When a navigation app like Google Maps calculates the best route, it's solving a complex system of equations based on traffic, distance, and speed. If a route can be found, the system is consistent. Similarly, ISRO scientists use consistent systems to calculate satellite orbits, ensuring they reach their destination correctly.

Key Vocabulary

Key Terms

SOLUTION: Values for variables that satisfy all equations | LINEAR EQUATION: An equation whose graph is a straight line | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | UNIQUE SOLUTION: Exactly one set of values for the variables | INFINITELY MANY SOLUTIONS: An unlimited number of sets of values for the variables

What's Next

What to Learn Next

Great job understanding consistent systems! Next, explore 'Inconsistent Systems of Linear Equations' to learn what happens when no solution exists. This will complete your understanding of all possible outcomes for systems of linear equations.