What is the Equation of a Line Passing Through the Origin?

S6-SA1-0256

Grade Level:

Class 10

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

Definition

What is it?

The equation of a line passing through the origin (0,0) describes all the points (x,y) that lie on that specific straight line. This line always goes through the point where the X-axis and Y-axis meet, like the center of a graph paper. Its general form is y = mx, where 'm' is the slope or gradient of the line.

Simple Example

Quick Example

Imagine you are buying samosas, and each samosa costs Rs. 10. If you buy 0 samosas, you pay Rs. 0. If you buy 1 samosa, you pay Rs. 10. If you buy 2 samosas, you pay Rs. 20. This relationship (Cost = 10 * Number of Samosas) can be plotted as a straight line passing through the origin (0 samosas, 0 cost).

Worked Example

Step-by-Step

PROBLEM: Find the equation of a line passing through the origin and also through the point (2, 6).

STEP 1: Understand the general form. A line passing through the origin has the equation y = mx.

---STEP 2: We are given a point (x, y) = (2, 6) that lies on the line. We need to find 'm', the slope.

---STEP 3: Substitute the coordinates of the given point into the general equation: 6 = m * 2.

---STEP 4: Solve for 'm'. Divide both sides by 2: m = 6 / 2.

---STEP 5: Calculate the value of m: m = 3.

---STEP 6: Now that we have 'm', substitute it back into the general equation y = mx.

---STEP 7: The equation of the line is y = 3x.

ANSWER: The equation of the line is y = 3x.

Why It Matters

Understanding lines through the origin is crucial in fields like AI/ML for linear regression models, in Physics for understanding direct proportionality (like Force = mass * acceleration), and in Engineering for designing systems where output is directly proportional to input. This concept forms the base for many advanced calculations, helping engineers design bridges or scientists predict chemical reactions.

Common Mistakes

MISTAKE: Forgetting that 'c' (the y-intercept) is zero for lines passing through the origin, and using y = mx + c. | CORRECTION: Always remember that if a line passes through (0,0), its y-intercept 'c' is 0, simplifying the equation to y = mx.

MISTAKE: Confusing 'm' (slope) with the coordinates 'x' or 'y'. | CORRECTION: 'm' is the slope, representing how steep the line is. 'x' and 'y' are the coordinates of any point on the line. First, find 'm' using a known point, then use it in y = mx.

MISTAKE: Incorrectly calculating the slope 'm' when given a point. For example, if (3,9) is given, some might say m = 3 instead of m = 9/3. | CORRECTION: To find 'm' from a point (x,y) on the line, always use the formula m = y/x.

Practice Questions

Try It Yourself

QUESTION: What is the equation of a line passing through the origin and the point (5, 15)? | ANSWER: y = 3x

QUESTION: A mobile data plan charges Rs. 20 per GB. If the cost (y) is directly proportional to the data used (x), write the equation of the line representing this cost. | ANSWER: y = 20x

QUESTION: A line passes through the origin and has a slope of -2. What is its equation? Also, find the y-coordinate when x = 4. | ANSWER: Equation: y = -2x. Y-coordinate when x=4: -8

MCQ

Quick Quiz

Which of the following equations represents a line passing through the origin?

y = 2x + 5

y = x - 3

y = 4x

y = -x + 1

The Correct Answer Is:

A line passing through the origin has the form y = mx, where 'c' (the y-intercept) is zero. Only y = 4x fits this form, as when x=0, y=0.

Real World Connection

In the Real World

In India, many small businesses use direct proportionality. For example, a street vendor selling chai might calculate daily earnings (y) based on the number of cups sold (x). If each cup costs Rs. 10, the equation is y = 10x, a line passing through the origin. This simple concept helps them track sales and plan their day.

Key Vocabulary

Key Terms

ORIGIN: The point (0,0) where the X and Y axes intersect | SLOPE (m): A measure of the steepness of a line; how much y changes for a unit change in x | LINEAR EQUATION: An equation whose graph is a straight line | COORDINATES (x,y): A pair of numbers that locate a point on a graph

What's Next

What to Learn Next

Great job understanding lines through the origin! Next, you should learn about the 'Equation of a Line in Slope-Intercept Form (y = mx + c)'. This will help you understand lines that don't necessarily pass through the origin, where 'c' becomes important.