Syllabus Unit 7 – Coordinate Geometry Straight Lines Various Forms of Equation of a Line

Various Forms of Equation of a Line

Master the various forms of the equation of a line including slope-intercept, point-slope, and intercept forms for CBSE Class 11 Applied Mathematics.

Slope-Intercept Form

When a line has a slope m and makes an intercept c on the y-axis, its equation is expressed as y = mx + c. This form is essential for identifying the steepness and the starting point of a linear function on a coordinate plane. For example, a line with slope 2 and y-intercept 3 is y = 2x + 3.

y = mx + c
Example: Solved Example: Find the equation of a line with slope 3 and y-intercept -2.
Show Step-by-Step Solution

Step 1: Identify m = 3 and c = -2.
Step 2: Substitute into y = mx + c.
Step 3: y = 3x + (-2).
Answer: y = 3x - 2.

Point-Slope Form

The point-slope form is used when the slope m of a line and a specific point (x₁, y₁) lying on the line are known. This form allows us to derive the equation of any non-vertical line given these two pieces of information. It serves as the foundation for deriving other forms of linear equations.

y - y₁ = m(x - x₁)
Example: Solved Example: Find the equation of a line passing through (2, 5) with slope 4.
Show Step-by-Step Solution

Step 1: Identify x₁ = 2, y₁ = 5, m = 4.
Step 2: Substitute into y - 5 = 4(x - 2).
Step 3: Simplify to y - 5 = 4x - 8.
Answer: y = 4x - 3.

Intercept Form

When a line makes non-zero intercepts a and b on the x-axis and y-axis respectively, its equation is given by the intercept form. This is particularly useful in business mathematics for determining break-even points or resource allocation limits. The intercepts represent the points (a, 0) and (0, b).

x/a + y/b = 1
Example: Solved Example: Find the equation of a line with x-intercept 4 and y-intercept 6.
Show Step-by-Step Solution

Step 1: Identify a = 4, b = 6.
Step 2: Substitute into x/4 + y/6 = 1.
Step 3: Multiply by 12 to clear denominators: 3x + 2y = 12.
Answer: 3x + 2y - 12 = 0.