Equation of a Line (Introduction)
The equation of a line is an algebraic expression that describes all the points lying on that line. The most common form is y = mx + c, where m is the slope and c is the y-intercept.
This concept bridges algebra and geometry — every linear equation in two variables represents a straight line, and every straight line can be described by a linear equation.
In Class 9, you learn the basics: slope, y-intercept, and how to write and graph equations of lines.
What is Equation of a Line (Introduction)?
Definition: The equation of a straight line in the Cartesian plane is a linear equation in x and y of the form:
y = mx + c
Where:
- m = slope (gradient) of the line = rise/run = (y₂−y₁)/(x₂−x₁)
- c = y-intercept (where the line crosses the y-axis)
General form: ax + by + c = 0, where a, b, c are constants and a, b are not both zero.
Equation of a Line (Introduction) Formula
Key formulas:
- Slope: m = (y₂ − y₁)/(x₂ − x₁)
- Slope-intercept form: y = mx + c
- Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)
- Horizontal line: y = k (slope = 0)
- Vertical line: x = k (slope undefined)
Properties of slope:
- Positive slope → line goes upward (left to right)
- Negative slope → line goes downward
- Zero slope → horizontal line
- Undefined slope → vertical line
- Parallel lines have equal slopes
- Perpendicular lines: m₁ × m₂ = −1
Solved Examples
Example 1: Example 1: Finding slope from two points
Problem: Find the slope of the line passing through (2, 3) and (5, 9).
Solution:
- m = (y₂ − y₁)/(x₂ − x₁) = (9 − 3)/(5 − 2) = 6/3 = 2
Answer: Slope = 2.
Example 2: Example 2: Writing equation from slope and intercept
Problem: Write the equation of a line with slope 3 and y-intercept −2.
Solution:
- y = mx + c = 3x + (−2) = 3x − 2
Answer: y = 3x − 2.
Example 3: Example 3: Finding y-intercept from equation
Problem: Find the slope and y-intercept of 2x + 3y = 12.
Solution:
- Rearrange: 3y = −2x + 12 → y = (−2/3)x + 4
- Slope = −2/3, y-intercept = 4
Answer: m = −2/3, c = 4.
Example 4: Example 4: Horizontal and vertical lines
Problem: Write equations for (a) a horizontal line through (3, 5) and (b) a vertical line through (−2, 7).
Solution:
- (a) Horizontal line: y = 5
- (b) Vertical line: x = −2
Answer: (a) y = 5, (b) x = −2.
Example 5: Example 5: Line through origin
Problem: A line passes through the origin and (4, 8). Find its equation.
Solution:
- m = (8−0)/(4−0) = 2. Since it passes through origin, c = 0.
- y = 2x
Answer: y = 2x.
Example 6: Example 6: Checking if point lies on line
Problem: Does the point (3, 7) lie on the line y = 2x + 1?
Solution:
- Substitute x = 3: y = 2(3) + 1 = 7
- The point (3, 7) satisfies the equation.
Answer: Yes, (3, 7) lies on y = 2x + 1.
Example 7: Example 7: Parallel lines
Problem: Are the lines y = 3x + 5 and y = 3x − 2 parallel?
Solution:
- Both have slope m = 3.
- Lines with equal slopes are parallel.
Answer: Yes, they are parallel.
Example 8: Example 8: Finding equation from two points
Problem: Find the equation of the line through (1, 2) and (3, 8).
Solution:
- m = (8−2)/(3−1) = 6/2 = 3
- Using y − y₁ = m(x − x₁): y − 2 = 3(x − 1)
- y = 3x − 3 + 2 = 3x − 1
Answer: y = 3x − 1.
Example 9: Example 9: Finding x-intercept
Problem: Find the x-intercept of y = 4x − 8.
Solution:
- At x-intercept, y = 0: 0 = 4x − 8 → x = 2
Answer: x-intercept = (2, 0).
Example 10: Example 10: Perpendicular lines
Problem: Line L₁ has slope 2. What is the slope of a line perpendicular to it?
Solution:
- For perpendicular lines: m₁ × m₂ = −1
- 2 × m₂ = −1 → m₂ = −1/2
Answer: Slope = −1/2.
Real-World Applications
Applications:
- Physics: Velocity-time graphs, Ohm's law (V = IR).
- Economics: Supply and demand curves, cost functions.
- Geography: Elevation profiles, distance calculations.
- Engineering: Linear relationships in design and analysis.
Key Points to Remember
- Every linear equation ax + by + c = 0 represents a straight line.
- y = mx + c is the slope-intercept form.
- Slope m = (y₂ − y₁)/(x₂ − x₁).
- Positive slope → rising line. Negative slope → falling line.
- Horizontal line: y = k (m = 0). Vertical line: x = k (m undefined).
- Parallel lines have equal slopes.
- Perpendicular lines: product of slopes = −1.
- y-intercept is found by setting x = 0. x-intercept by setting y = 0.
Practice Problems
- Find the slope of the line through (−1, 4) and (3, −2).
- Write the equation of a line with slope −2 and y-intercept 5.
- Convert 3x − 4y = 20 to slope-intercept form.
- Find the equation of the line through (2, 5) and (6, 13).
- Does (−1, 3) lie on y = −2x + 1?
- Find both intercepts of the line 2x + 5y = 10.
- Are y = −4x + 3 and y = (1/4)x − 1 perpendicular?
Frequently Asked Questions
Q1. What is the slope of a line?
The slope (m) measures steepness: m = rise/run = (y₂−y₁)/(x₂−x₁). It tells how much y changes for each unit change in x.
Q2. What is the y-intercept?
The y-intercept (c) is the y-coordinate where the line crosses the y-axis. At this point, x = 0.
Q3. Can every line be written as y = mx + c?
No. Vertical lines (x = k) cannot be written in this form because their slope is undefined.
Q4. How do you know if two lines are parallel?
Two lines are parallel if and only if they have the same slope but different y-intercepts.
Q5. What does a negative slope mean?
The line goes downward from left to right. As x increases, y decreases.
Q6. What is the equation of a line through the origin?
y = mx (c = 0). The y-intercept is 0 because the line passes through (0, 0).
Related Topics
- Graph of Linear Equation in Two Variables
- Cartesian Plane
- Distance Formula
- Introduction to Coordinate Geometry
- Plotting Points in Four Quadrants
- Abscissa and Ordinate
- Section Formula
- Midpoint Formula
- Area of Triangle Using Coordinates
- Centroid of a Triangle
- Collinearity Using Distance Formula
- External Division (Section Formula)
- Coordinate Geometry Word Problems
- Quadrilateral Using Coordinate Geometry
