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What is the System of Equations with Infinitely Many Solutions?
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 infinitely many solutions means that all the equations in the system are essentially the same or represent the same line/plane. This means there are countless sets of values for the variables that satisfy all equations simultaneously, like having two identical maps of the same route.
Simple Example
Quick Example
Imagine you have two statements about the price of a chai and a samosa. Statement 1: 'One chai and one samosa cost 30 rupees.' Statement 2: 'Two chai and two samosas cost 60 rupees.' Notice that the second statement is just double the first one. Any combination of chai and samosa prices that satisfies the first statement (like chai=20, samosa=10) will also satisfy the second, and there are many such combinations.
Worked Example
Step-by-Step
Let's solve the system:
Equation 1: x + 2y = 5
Equation 2: 2x + 4y = 10
Step 1: Look at Equation 1: x + 2y = 5.
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Step 2: Look at Equation 2: 2x + 4y = 10. Can we simplify it?
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Step 3: Divide all terms in Equation 2 by 2: (2x)/2 + (4y)/2 = 10/2. This gives x + 2y = 5.
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Step 4: Notice that after simplifying, Equation 2 (x + 2y = 5) is exactly the same as Equation 1 (x + 2y = 5).
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Step 5: Since both equations are identical, any pair of (x, y) values that satisfies one equation will satisfy the other. For example, if x=1, then 1 + 2y = 5 => 2y = 4 => y=2. So (1, 2) is a solution. If x=3, then 3 + 2y = 5 => 2y = 2 => y=1. So (3, 1) is another solution.
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Step 6: We can express the solutions by letting one variable be 'k'. From x + 2y = 5, we can write x = 5 - 2y. So, the solutions are of the form (5 - 2k, k), where 'k' can be any real number.
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Answer: The system has infinitely many solutions, represented by (5 - 2k, k).
Why It Matters
Understanding systems with infinitely many solutions is crucial in fields like AI/ML for optimizing algorithms and in engineering for designing stable systems. For example, in climate science, if different models predict the same outcome based on varied inputs, it might indicate infinitely many ways to reach that outcome, helping engineers build more robust systems or AI models.
Common Mistakes
MISTAKE: Assuming there are no solutions when equations look similar but are not exactly the same after simplification (e.g., x+y=5 and x+y=6). | CORRECTION: Always simplify equations fully and compare coefficients and constants. If coefficients are proportional but constants are not, there are no solutions.
MISTAKE: Only finding one or two solutions and stopping, thinking that's all there is. | CORRECTION: If the equations simplify to be identical, remember to state that there are 'infinitely many solutions' and often express them in terms of a parameter (like 'k').
MISTAKE: Not checking if all equations are consistent. | CORRECTION: Ensure that when you simplify or manipulate the equations, they truly become identical. If even one equation doesn't match the others, it might lead to no solution or a unique solution, not infinite.
Practice Questions
Try It Yourself
QUESTION: Does the system below have infinitely many solutions?
Equation 1: 3x - y = 6
Equation 2: 6x - 2y = 12 | ANSWER: Yes
QUESTION: Find the form of the solutions for the system:
Equation 1: 4x + 8y = 20
Equation 2: x + 2y = 5 | ANSWER: (5 - 2k, k)
QUESTION: For what value of 'c' will the following system have infinitely many solutions?
Equation 1: 2x + 3y = 7
Equation 2: 4x + 6y = c | ANSWER: c = 14
MCQ
Quick Quiz
Which of the following systems of equations has infinitely many solutions?
x + y = 4; x - y = 2
2x + 3y = 6; 4x + 6y = 12
x + 2y = 5; x + 2y = 7
3x - y = 9; x + y = 1
The Correct Answer Is:
B
Option B's second equation (4x + 6y = 12) is simply double the first equation (2x + 3y = 6). This means both equations represent the same line, leading to infinitely many solutions. Other options either have a unique solution or no solution.
Real World Connection
In the Real World
Imagine you're developing an app to manage delivery routes for a local kirana store. If two different equations describing the shortest path between two points turn out to be identical (meaning there are multiple ways to describe the same shortest path), it means your optimization algorithm has found infinitely many 'best' ways to represent that path. This is important in logistics and supply chain management, like for Zepto or Swiggy deliveries, to ensure flexibility and efficiency.
Key Vocabulary
Key Terms
SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | INFINITELY MANY SOLUTIONS: Countless solutions that satisfy all equations in a system | CONSISTENT SYSTEM: A system that has at least one solution (could be one or infinite) | DEPENDENT EQUATIONS: Equations that are essentially the same, one can be derived from the other
What's Next
What to Learn Next
Great job understanding infinitely many solutions! Next, you should explore 'Systems of Equations with No Solutions'. This will help you understand all possible outcomes when solving equations and prepare you for more complex problems in higher classes.
