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What is a Unique Solution for Systems of Equations (Graphically)?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A unique solution for a system of linear equations means there is only one specific pair of values (x, y) that satisfies all the equations at the same time. Graphically, this happens when the lines representing the equations intersect at exactly one point.
Simple Example
Quick Example
Imagine two friends, Rohan and Priya, selling samosas. Rohan charges Rs. 10 for each samosa and has a fixed delivery charge of Rs. 20. Priya charges Rs. 15 per samosa with no delivery charge. If we want to find out for how many samosas their total cost will be exactly the same, there will be only one specific number of samosas where their costs meet. This single meeting point is a unique solution.
Worked Example
Step-by-Step
Let's find the unique solution for the system of equations graphically:
Equation 1: y = x + 1
Equation 2: y = -x + 5
Step 1: For Equation 1 (y = x + 1), find two points. If x=0, y=1 (Point A: 0,1). If x=2, y=3 (Point B: 2,3).
---Step 2: Plot Points A and B on a graph paper and draw a straight line passing through them.
---Step 3: For Equation 2 (y = -x + 5), find two points. If x=0, y=5 (Point C: 0,5). If x=2, y=3 (Point D: 2,3).
---Step 4: Plot Points C and D on the same graph paper and draw a straight line passing through them.
---Step 5: Observe where the two lines intersect. They cross each other at exactly one point.
---Step 6: Read the coordinates of this intersection point from the graph. The lines intersect at (2, 3).
---Step 7: This intersection point (2, 3) is the unique solution. It means when x=2 and y=3, both equations are true.
Answer: The unique solution is (2, 3).
Why It Matters
Understanding unique solutions is super important in fields like AI/ML and Data Science, where computers solve many equations to make predictions or understand patterns. Engineers use this to design structures, and economists use it to find balance points in markets. It helps us find single, clear answers to complex problems.
Common Mistakes
MISTAKE: Thinking that if lines are parallel, they have a unique solution. | CORRECTION: Parallel lines never intersect, so they have no solution, not a unique one.
MISTAKE: Drawing lines inaccurately, leading to a wrong intersection point. | CORRECTION: Always plot at least two points for each line carefully and use a ruler to draw straight lines to ensure accuracy.
MISTAKE: Confusing a unique solution with 'many solutions' (infinite solutions). | CORRECTION: A unique solution means the lines cross at ONE point. 'Many solutions' happens when the two lines are exactly the same (overlap completely).
Practice Questions
Try It Yourself
QUESTION: Graphically find the unique solution for: y = 2x and y = x + 1. | ANSWER: (1, 2)
QUESTION: Plot the lines for x + y = 5 and x - y = 1. What is their unique solution? | ANSWER: (3, 2)
QUESTION: A mobile data plan costs Rs. 100 plus Rs. 5 per GB used. Another plan costs Rs. 200 plus Rs. 2 per GB used. Write two equations (let C be cost, G be GB) and graphically find for what GB usage their costs are the same (unique solution). | ANSWER: Equations: C = 5G + 100, C = 2G + 200. Unique solution: G = 33.33 (approx), C = 266.67 (approx).
MCQ
Quick Quiz
Which of these describes a unique solution graphically?
The two lines are parallel.
The two lines overlap completely.
The two lines intersect at exactly one point.
The two lines do not exist.
The Correct Answer Is:
C
A unique solution means there is only one answer, which graphically translates to the lines crossing at exactly one point. Parallel lines have no solution, and overlapping lines have infinite solutions.
Real World Connection
In the Real World
Think about navigation apps like Google Maps or Ola/Uber. When you search for the fastest route, the app's algorithms are solving many equations simultaneously to find the single best path. This 'best path' is like a unique solution, where various factors (distance, traffic, road closures) meet to give one optimal outcome.
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 | INTERSECTION POINT: The point where two lines cross each other | COORDINATES: A pair of numbers (x, y) that show the exact position of a point on a graph.
What's Next
What to Learn Next
Great job understanding unique solutions! Next, you should explore what happens when systems of equations have 'no solution' (parallel lines) or 'infinite solutions' (overlapping lines). This will complete your understanding of all possible outcomes for linear systems.
