What is the Equation of a Line? | Simple Explanation for Class 10

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What is the Equation of a Line Passing Through Two Given Points?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of a line passing through two given points is a mathematical formula that helps us describe the exact path of a straight line when we know the coordinates of any two points it crosses. It's like finding the unique address of a road if you know two specific landmarks on it.

Simple Example

Quick Example

Imagine you are tracking an auto-rickshaw on a map. At 10:00 AM, it's at point A (2 km East, 3 km North from your home). At 10:15 AM, it's at point B (6 km East, 7 km North). The equation of the line tells you the exact straight path the auto took between these two points, helping you predict its location at any time in between.

Worked Example

Step-by-Step

Let's find the equation of a line passing through points P(3, 5) and Q(7, 13).

Step 1: Identify the coordinates of the two points. Let (x1, y1) = (3, 5) and (x2, y2) = (7, 13).

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Step 2: Calculate the slope (m) of the line using the formula m = (y2 - y1) / (x2 - x1).
m = (13 - 5) / (7 - 3)
m = 8 / 4
m = 2

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Step 3: Use the point-slope form of the line equation: y - y1 = m(x - x1). You can use either point; let's use (3, 5).
y - 5 = 2(x - 3)

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Step 4: Simplify the equation to the standard form y = mx + c.
y - 5 = 2x - 6
y = 2x - 6 + 5
y = 2x - 1

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Step 5: Verify by plugging in the second point (7, 13) into the equation.
13 = 2(7) - 1
13 = 14 - 1
13 = 13 (This is correct!)

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Answer: The equation of the line is y = 2x - 1.

Why It Matters

Understanding line equations is crucial in fields like AI/ML for drawing decision boundaries, in Physics for plotting motion, and in Engineering for designing structures. Future engineers, data scientists, and even doctors using medical imaging rely on these basic math principles.

Common Mistakes

MISTAKE: Swapping x and y coordinates when calculating slope, like (x2 - x1) / (y2 - y1) | CORRECTION: Always remember slope is 'rise over run', so it's (change in y) / (change in x), or (y2 - y1) / (x2 - x1).

MISTAKE: Incorrectly applying negative signs, especially when one or more coordinates are negative. For example, (5 - (-2)) becoming (5 - 2) | CORRECTION: Be very careful with subtraction of negative numbers. Remember that subtracting a negative is the same as adding a positive (e.g., 5 - (-2) = 5 + 2 = 7).

MISTAKE: Using the wrong point in the point-slope form, or mixing up x1 with x2, or y1 with y2. | CORRECTION: Once you pick a point (x1, y1) to use in y - y1 = m(x - x1), stick with its coordinates. Don't mix parts of one point with another.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the line passing through points A(1, 2) and B(3, 8). | ANSWER: y = 3x - 1

QUESTION: A line passes through P(-2, 5) and Q(4, -1). What is its equation? | ANSWER: y = -x + 3

QUESTION: The cost of 5 samosas is Rs 75 and the cost of 10 samosas is Rs 125. Assuming a linear relationship between the number of samosas and their cost, find the equation that represents this relationship. Let x be the number of samosas and y be the total cost. | ANSWER: y = 10x + 25

MCQ

Quick Quiz

Which of these equations represents a line passing through (0, 0) and (2, 4)?

y = x + 2

y = 2x

y = x - 2

y = 4x - 2

The Correct Answer Is:

For points (0,0) and (2,4), the slope m = (4-0)/(2-0) = 4/2 = 2. Using y - y1 = m(x - x1) with (0,0): y - 0 = 2(x - 0) which simplifies to y = 2x.

Real World Connection

In the Real World

Delivery apps like Zomato or Swiggy use this concept to calculate the shortest straight-line distance between the restaurant and your home, helping estimate delivery times. Urban planners use it to design road networks, ensuring efficient routes between two important locations in a city.

Key Vocabulary

Key Terms

COORDINATES: A pair of numbers (x, y) that show the exact position of a point on a graph | SLOPE: The steepness of a line, calculated as 'rise over run' | Y-INTERCEPT: The point where a line crosses the Y-axis | LINEAR EQUATION: An equation whose graph is a straight line, usually in the form y = mx + c

What's Next

What to Learn Next

Next, you can explore how to find the equation of a line when you only know its slope and one point, or when you know its slope and y-intercept. These concepts build directly on what you've learned here and are super useful!