What is the Relationship Between Solutions and Intersections?

S3-SA5-0182

Grade Level:

Class 10

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

Definition

What is it?

The relationship between solutions and intersections is that a 'solution' to a system of equations is precisely where their graphs 'intersect'. When two lines or curves cross each other on a graph, the point(s) where they meet represent the values that satisfy all equations in the system simultaneously.

Simple Example

Quick Example

Imagine two friends, Rohan and Priya, selling 'chai' at different prices. Rohan sells chai for Rs. 10 per cup and has a fixed daily expense of Rs. 50. Priya sells chai for Rs. 12 per cup and has a fixed daily expense of Rs. 30. If we want to find out when their total daily costs are the same, we're looking for the 'intersection' point of their cost equations, which gives us the 'solution'.

Worked Example

Step-by-Step

Let's find the solution for the system of equations: y = 2x + 1 and y = -x + 4.

1. We want to find the point (x, y) where both equations are true. This means the 'y' values must be equal.
---2. Set the two expressions for 'y' equal to each other: 2x + 1 = -x + 4.
---3. Now, solve for 'x'. Add 'x' to both sides: 2x + x + 1 = 4, which simplifies to 3x + 1 = 4.
---4. Subtract '1' from both sides: 3x = 4 - 1, which means 3x = 3.
---5. Divide by '3': x = 3 / 3, so x = 1.
---6. Now that we have 'x', substitute x = 1 into either of the original equations to find 'y'. Let's use y = 2x + 1.
---7. y = 2(1) + 1 = 2 + 1 = 3.
---8. So, the solution is (x, y) = (1, 3). This is the point where the graphs of these two lines intersect.

Answer: The solution and intersection point is (1, 3).

Why It Matters

Understanding solutions and intersections is crucial in many fields. In AI/ML, it helps algorithms find optimal points for data analysis. Data scientists use it to identify trends and make predictions, while engineers use it to design systems where different components interact optimally, like finding the best path for a delivery drone or balancing loads in a bridge.

Common Mistakes

MISTAKE: Finding only the 'x' value and forgetting to find the 'y' value. | CORRECTION: Remember that an intersection point is a coordinate (x, y), so both values are needed. Always substitute the found 'x' back into one of the original equations to get 'y'.

MISTAKE: Making calculation errors when rearranging equations to solve for 'x' or 'y'. | CORRECTION: Double-check each step of your algebra. Pay close attention to signs (+/-) when moving terms across the equals sign.

MISTAKE: Assuming there's always one unique intersection point. | CORRECTION: For lines, there can be one intersection, no intersection (parallel lines), or infinitely many intersections (same line). For curves, there can be multiple intersection points.

Practice Questions

Try It Yourself

QUESTION: Find the solution to the system: y = 3x - 2 and y = x + 4. | ANSWER: x = 3, y = 7 (or (3, 7))

QUESTION: If the cost of a 'samosa' (C) depends on its size (S) as C = 5S + 10, and the selling price (P) is P = 7S + 4, find the size (S) where cost and selling price are equal. What is that price? | ANSWER: S = 3, P = 31

QUESTION: Two mobile data plans are offered. Plan A: Rs. 100 fixed charge + Rs. 5 per GB. Plan B: Rs. 50 fixed charge + Rs. 10 per GB. At what data usage (in GB) will both plans cost the same? Write the equations and solve. | ANSWER: Equations: C_A = 5x + 100, C_B = 10x + 50. Solution: x = 10 GB. Cost = Rs. 150.

MCQ

Quick Quiz

What does the intersection point of two linear equations represent?

The sum of their slopes

The value that satisfies only one equation

The point where both equations are simultaneously true

The product of their y-intercepts

The Correct Answer Is:

The intersection point is where the graphs of the equations meet. This means that at this specific point, the x and y values satisfy both equations at the same time, making them simultaneously true.

Real World Connection

In the Real World

Think about ride-sharing apps like Ola or Uber. They use algorithms that find the 'intersection' of demand for rides and available drivers in a certain area. This helps them set dynamic pricing, ensuring enough cars are available when and where people need them, balancing driver earnings and passenger fares.

Key Vocabulary

Key Terms

SOLUTION: The set of values that makes an equation or system of equations true. | INTERSECTION: The point(s) where two or more graphs meet or cross. | SYSTEM OF EQUATIONS: A collection of two or more equations with the same set of variables. | SIMULTANEOUSLY: Happening or being true at the same time.

What's Next

What to Learn Next

Now that you understand solutions and intersections for linear equations, you can explore 'Solving Systems of Equations by Substitution and Elimination'. This will teach you more methods to find these important points without always needing to graph, which is super useful for more complex problems!