How to Find the Equation of a Line from Two Points?
The concept of Equation of a Line plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Equation of a Line?
An Equation of a Line is an algebraic expression representing every point that lies on a straight line in a plane. You’ll find this concept applied in areas such as coordinate geometry, linear algebra, and graphing in everyday calculations and competitive exams.
Key Formula for Equation of a Line
Here’s the standard formula: \( ax + by + c = 0 \), where a, b, and c are real numbers, and x, y are variables representing any point on the line.
The slope-intercept form is: \( y = mx + c \)
Point-slope form: \( y - y_1 = m(x - x_1) \)
Two-point form: \( \frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1} \)
Cross-Disciplinary Usage
Equation of a line is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions involving motion, graphs, and optimization.
Step-by-Step Illustration
Let’s find the equation of a line passing through points (2, 3) and (5, 11):
1. Find the slope (m):2. Use the point-slope form with one point (2, 3):
3. Expand:
4. Rearranged in standard form:
This is the equation of the line passing through (2, 3) and (5, 11).
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with equation of a line. When you have two points, you can directly substitute values into the two-point formula:
- Write the formula: \( y - y_1 = \frac{y_2-y_1}{x_2-x_1} (x-x_1) \)
- Substitute all values in one go without calculating slope separately.
- This reduces errors and saves time during exams!
Tricks like this are especially useful in competitive exams like NTSE, Olympiad, and JEE. Vedantu’s live sessions feature more such Vedic maths techniques to build speed and accuracy.
Try These Yourself
- Write the equation of a line through (1, 2) with slope 3.
- Find the equation of a line passing through points (-1, 4) and (2, 10).
- If a line passes through (0, 5) and has slope -2, what is its equation?
- Is y = 2x + 3 same as 2x - y + 3 = 0? Explain.
Frequent Errors and Misunderstandings
- Swapping x and y values when finding slope.
- Forgetting to rearrange equation into standard form.
- Confusing slope (m) with intercept (c).
- Not checking if points satisfy the final equation.
Relation to Other Concepts
The idea of equation of a line connects closely with topics such as Slope of Line and Coordinate Geometry. Mastering this helps with understanding tangent and normal equations, distances, and areas in advanced mathematics.
Classroom Tip
A quick way to remember line equations is: "Slope first, then intercept!" Always check which information is given (points, slope, or intercept) and use the matching formula form. Vedantu’s teachers love to use colored chalk or visuals to reinforce these patterns during live classes.
We explored Equation of a Line—from definition, formula, examples, common mistakes, and connections to key maths concepts. Continue practicing with Vedantu to become confident in solving all types of line equation problems.
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FAQs on Equation of a Line – Definition, Forms, and Examples
1. What is the equation of a line in Maths?
The equation of a line represents the algebraic relationship between the x and y coordinates of all points on a straight line. The most common form is the general equation: Ax + By + C = 0, where A, B, and C are constants. Other forms, like the slope-intercept form (y = mx + c) and the point-slope form (y - y₁ = m(x - x₁)), offer different ways to represent the same line, each useful in specific situations.
2. How do you find the equation of a line given two points?
To find the equation of a line passing through two points (x₁, y₁) and (x₂, y₂):
- Calculate the slope (m): m = (y₂ - y₁) / (x₂ - x₁)
- Use the point-slope form: y - y₁ = m(x - x₁), substituting the slope and one of the points.
- Simplify the equation to your preferred form (e.g., slope-intercept or general form).
3. What is the standard form of a line's equation?
The standard form is often considered to be the general form: Ax + By + C = 0, where A, B, and C are constants. However, the 'standard' form can also refer to the slope-intercept form (y = mx + c) depending on the context. Both forms are widely used and equally valid.
4. What is the equation of a line in 3D space?
In three dimensions, a line is typically represented using parametric equations. These equations define the coordinates (x, y, z) of any point on the line in terms of a parameter, usually 't'. A common representation is: x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where (x₀, y₀, z₀) is a point on the line and (a, b, c) is the direction vector.
5. How to find a line perpendicular to another given line?
If a line has a slope 'm', any line perpendicular to it will have a slope of -1/m (the negative reciprocal). Use this perpendicular slope and either a point on the perpendicular line or the point-slope form to find the equation of the perpendicular line.
6. What are the different forms of the equation of a line?
Common forms include: General form (Ax + By + C = 0), Slope-intercept form (y = mx + c), Point-slope form (y - y₁ = m(x - x₁)), Two-point form (derived from the slope and two points), and Intercept form (x/a + y/b = 1). The choice of form depends on the given information and the desired outcome.
7. How do I find the x and y intercepts of a line?
To find the x-intercept, set y = 0 in the line's equation and solve for x. To find the y-intercept, set x = 0 and solve for y. The intercepts are the points where the line crosses the x-axis and y-axis, respectively.
8. What are parallel lines?
Parallel lines are lines that never intersect. In two dimensions, they have the same slope. In 3D space, their direction vectors are proportional.
9. How can I determine if two lines are parallel or perpendicular?
Two lines are parallel if their slopes are equal (or direction vectors are proportional in 3D). They are perpendicular if the product of their slopes is -1 (or their direction vectors are orthogonal in 3D).
10. What is the equation of a line parallel to the x-axis or y-axis?
A line parallel to the x-axis has the equation y = k, where 'k' is a constant representing the y-coordinate. A line parallel to the y-axis has the equation x = k, where 'k' is a constant representing the x-coordinate.
11. How is the equation of a line used in real-world applications?
Line equations are fundamental in many fields. They are used in:
- Computer graphics (representing lines and shapes)
- Physics (modeling linear motion)
- Engineering (designing structures and systems)
- Economics (linear relationships between variables)
- Cartography (representing geographical features)
