Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS
Introduction
The standard form of a straight line equation is one of the fundamental representations of a line in coordinate geometry. It is widely used in CBSE Class 11 Mathematics and JEE Mains & Advanced to analyze linear equations and their properties.
Table of Contents
Slope Intercept Form of Standard Form of a Straight Line
As we know, the standard form of the equation of a straight line is:
Ax + By + C = 0
Rearranging the terms as:
By = -Ax – C
⇒y = (-A/B)x + (-C/B)
This is of the form y = mx + c
Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.
Solved Examples
Examples 1: Find the slope and the y-intercept of the given equation, 2x + 5y + 1 = 0.
Solution:
Given: equation of line = 2x + 5y + 1 = 0
Find: slope and y-intercept
So the given equation can be written as
y = (-1 – 2x)/5
y = -2x/5 – 1/5…(1)
As we know that the slope-intercept form is
y = mx + c…(2)
On comparing eq(1) and (2) we get
m = -2/5 and c = -1/5
Hence, the slope is -2/5 and y-intercept is -1/5
Example 2: Find the slope and the y-intercept of the given equation, 3x + 6y – 9 = 0.
Solution:
Given: equation of line = 3x + 6y – 9 = 0
Find: slope and y-intercept
3x +6y-9 = 0
6y = -3x + 9
y = -1/2(x) + 3/2
Comparing this with the slope-intercept form, we find:
Slope ? = -1/2
Y-intercept ? = 3/2
So, the slope of the given equation is -1/2 and y – intercept is 3/2
FAQs on Standard Form of a Straight Line
What is the standard form of a straight line?
The standard form of a straight line is Ax+By=C, where A, B, and C are constants, and A and B are not both zero.
Why is the standard form useful?
The standard form is useful because it provides a consistent way to represent and manipulate linear equations. It simplifies the process of finding intercepts, solving systems of equations, and analyzing linear relationships.
How do you convert a linear equation to standard form?
To convert a linear equation to standard form, rearrange the equation so that all terms involving variables x and y are on one side of the equation, with the constant term on the other side. Ensure that A, B, and C are integers, and A should be positive.
How can you find the x-intercept and y-intercept from the standard form?
To find the x-intercept, set y=0 and solve for x. To find the y-intercept, set x=0 and solve for y.
Can the coefficients A, B, and C be fractions or decimals?
While A, B, and C can technically be fractions or decimals, it is standard practice to represent them as integers. This can be achieved by multiplying through by a common denominator if necessary.
What is the slope of a line in standard form?
The slope of a line in standard form Ax+By=C is given by −A/B, provided B≠0.
How do you graph a line given its standard form equation?
To graph a line given its standard form equation, find the x-intercept and y-intercept by setting y=0 and x=0 respectively, then plot these intercepts and draw the line through them.
What is the relationship between standard form and slope-intercept form?
The slope-intercept form of a line is y=mx+b, where m is the slope and b is the y-intercept. You can convert from standard form to slope-intercept form by solving the standard form equation for y.
Can a vertical or horizontal line be represented in standard form?
Yes, a vertical line can be represented as Ax=C (since B=0), and a horizontal line can be represented as By=C (since A=0).
How do you determine if two lines are parallel or perpendicular from their standard form equations?
Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is −1. For lines in standard form Ax+By=C, calculate their slopes as −A/B to check for these conditions.
Practice Problems
- Find the slope of the line y = 5x + 2.
- Find the slope of the line which crosses the line at point (-2,6) and have an intercept of 3.
- What is the equation of the line whose angle of inclination is 45 degrees and x-intercept is -⅗?
- Write the equation of the line passing through the point (0, 0) with slope -4.
Key Takeaways
✅ The standard form of a straight-line equation is Ax+By+C=0
✅ It helps in easily identifying coefficients and solving equations
✅ Other forms (slope-intercept, point-slope, and intercept) can be converted to standard form
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