What is a Solution Set on a Graph?

S3-SA5-0133

Grade Level:

Class 10

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

Definition

What is it?

A solution set on a graph is the collection of all points (x, y) that satisfy a given equation or inequality. When you plot these points, they form a line, curve, or shaded region that visually represents all possible solutions.

Simple Example

Quick Example

Imagine you have a rule that says 'The number of samosas (y) must be exactly double the number of chai cups (x)'. If you plot points like (1 chai, 2 samosas), (2 chai, 4 samosas), etc., all these points form a line on a graph. This line is the solution set for the rule y = 2x.

Worked Example

Step-by-Step

Let's find the solution set for the equation y = x + 2 on a graph.

1. Choose a few simple values for x.

---

2. Calculate the corresponding y values using the equation.
If x = -1, y = -1 + 2 = 1. So, point is (-1, 1).
If x = 0, y = 0 + 2 = 2. So, point is (0, 2).
If x = 1, y = 1 + 2 = 3. So, point is (1, 3).
If x = 2, y = 2 + 2 = 4. So, point is (2, 4).

---

3. Plot these points on a coordinate plane.

---

4. Draw a straight line connecting these points.

---

5. This line is the graphical representation of the solution set for y = x + 2. Every point on this line, and only points on this line, will satisfy the equation.

Answer: The line passing through points like (-1,1), (0,2), (1,3), (2,4) is the solution set.

Why It Matters

Understanding solution sets on a graph is super important for fields like AI/ML and Data Science, where you often need to visualize data patterns and relationships. Engineers use them to design structures, and economists use them to predict market trends. It helps solve real-world problems by showing all possible answers visually.

Common Mistakes

MISTAKE: Only plotting two points for a linear equation and assuming the line is correct. | CORRECTION: Plot at least three points. If they don't form a straight line, you've made a calculation error for one of the points. This helps catch mistakes.

MISTAKE: Confusing the x and y coordinates when plotting points (e.g., plotting (2,3) as (3,2)). | CORRECTION: Always remember the order is (x-coordinate, y-coordinate). The x-value tells you how far horizontally from the origin, and the y-value tells you how far vertically.

MISTAKE: Not extending the line or curve beyond the plotted points, especially for equations that have infinite solutions. | CORRECTION: For linear equations, draw arrows at both ends of the line to show that the solution set continues infinitely in both directions.

Practice Questions

Try It Yourself

QUESTION: Plot the solution set for the equation y = 3x on a graph. List three points that are part of this solution set. | ANSWER: Points like (0,0), (1,3), (2,6) and a line passing through them.

QUESTION: A mobile data plan charges Rs 50 for activation plus Rs 10 per GB used. If 'y' is the total cost and 'x' is GB used, write an equation and plot its solution set. Is (5, 90) a part of this solution set? | ANSWER: Equation: y = 10x + 50. (5, 90) is NOT part of the solution set because 10(5) + 50 = 100, not 90.

QUESTION: The sum of two numbers is 7. If the first number is x and the second is y, plot the solution set for this relationship. Also, plot the solution set for y = x - 1 on the same graph. What is the common point in both solution sets? | ANSWER: Equation 1: x + y = 7. Equation 2: y = x - 1. The common point (intersection) is (4, 3).

MCQ

Quick Quiz

Which of the following points is part of the solution set for the equation y = 2x + 1?

(1, 2)

(2, 5)

(0, 0)

(-1, 0)

The Correct Answer Is:

For option B, if x=2, then y = 2(2) + 1 = 4 + 1 = 5. So, (2, 5) satisfies the equation. The other options do not satisfy the equation.

Real World Connection

In the Real World

Imagine a food delivery app like Swiggy or Zomato. They use solution sets on graphs to figure out the most efficient delivery routes. The graph might represent roads, and the solution set could be the shortest path connecting the restaurant to your home, considering traffic and distance. This ensures your biryani reaches you hot and fast!

Key Vocabulary

Key Terms

EQUATION: A mathematical statement showing two expressions are equal, usually with an '=' sign. | INEQUALITY: A mathematical statement showing that two expressions are not equal, using symbols like <, >, <=, >=. | COORDINATE PLANE: A 2D surface formed by two perpendicular number lines (x-axis and y-axis) used for plotting points. | ORIGIN: The point (0,0) where the x-axis and y-axis intersect on a coordinate plane. | LINEAR EQUATION: An equation whose graph is a straight line.

What's Next

What to Learn Next

Next, you can explore 'Solving Systems of Linear Equations Graphically'. This builds on solution sets by finding the point (or points) where the solution sets of two or more equations intersect, which is crucial for solving real-world problems.