How to Find the Slope and Intercept Easily
In this article, we’ll discuss the equation of a straight line formula, gradient of a straight line formula. We’ll also discuss some examples of how to find the equation of a straight line when Co-ordinates or gradients are given to us. There are n-number of ways to express an equation of a straight line and some are more general than others.
In order to master the techniques explained here, it's vital that you simply undertake many practice exercises so that they become second nature.
After reading this article, you will be able to solve:
How to find the equation of a line, given its gradient and its intercept on the y-axis;
How to find the equation of a straight line, given its gradient and one point lying on it;
How to find the equation of a line given two points lying on it;
Find the equation of a straight line in either of the forms y = mx + c or ax + by + c = 0.
Define Equation of a Straight Line
So, how do we define the equation of a straight line? When we started studying geometry first we studied point and then line. The very first intuition is, a line is a set of points which holds good in certain conditions. The definition of a line is, a straight line is the collection of points in between and extending beyond two points. In most geometrical perspectives, a line is an object that does not have formal properties other than it’s length, it’s single-dimensional. You may find the various equations of a straight line calculator on the internet.
General Equation of a Straight Line Formula
The general equation of straight line is
ax + by + c = 0
Here x and y are the coordinate axes and a, b ,c are the constants.
The Slope of a Straight Line
The slope of a straight line is also known as the gradient of a straight line. Actually, it is the tangent of an angle. An angle of the straight line from the positive direction of the x-axis. Usually, it is represented by m and its formula is m = tanፀ. Where ፀ (theta) is the angle of the straight line from the positive direction of the x-axis considered anticlockwise as mentioned in the following diagram.
There is one more formula for getting the slope of a line when two points are given, the line is passing through. Let (x1,y1) and (x2,y2) are the point and we are required to find the gradient then we’ll use the formula
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Slope Intercept Form of the Straight Line
It is the most used and easiest form of the straight line. The equation for slope-intercept form is y = mx + c, where x and y are axes, m is the slope whose formula is tanፀ and c is the intercept of line on the y-axis.
The Intercept Form of the Straight Line
The intercept form of a straight line can be written as
\[\frac{x}{a}+\frac{y}{b}=1\]
Here, a and y are coordinate axes and a, b are intercepted on the x and y-axis respectively.
The geometrical representation of intercept form is mentioned below:
The Normal Form of a Straight Line
The normal form of the straight line is
x cos α + y sin α = p
Here, x and y are coordinates, p is the length of the perpendicular from origin to the straight line and α is the angle between the positive x-axis and the perpendicular of the straight line from the origin. As mentioned below
One important point we need to keep in mind is p is always positive and measured away from the origin. α is a positive angle that is less than 360⁰ measured from the positive direction of the x-axis, Many students misunderstand this.
Slope One Point Form of the Straight Line
When the slope and a point through which the required straight line is passing, are given to us and we need to find the equation of the straight line the use the formula
y - y1 = m(x - x1)
Here, (x1,y1) is the point on the coordinate plane through which the line is passing and m is the slope.
Two-Point Form of the Straight Line
In the previous section, we have discussed the one point form of the straight line which is y - y1 = m(x - x1)
In this formula, m is the slope and we know that when two points are given then the slope can be computed as \[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] . We can substitute this slope formula in the previous equation which will give us
\[y-y_{1}=(\frac{y_{2}-y_{1}}{x_{2}-x_{1}})(x-x_{1})\]
Or, (y - y1)(y2 - y1) = (x - x1)(x2 - x1)
FAQs on Equation of a Straight Line Explained
1. What is the fundamental concept explained by the equation of a straight line?
The fundamental concept behind the equation of a straight line is that it represents the algebraic relationship between all the points (x, y) that lie on that line. Any point that satisfies the equation is on the line, and any point on the line will satisfy its equation. It essentially provides a rule that connects the x-coordinate and y-coordinate for every point on that specific line.
2. What are the different forms used to represent the equation of a straight line?
The equation of a straight line can be represented in several forms, each useful in different situations. The primary forms as per the CBSE syllabus are:
- Slope-Intercept Form: y = mx + c, where 'm' is the slope and 'c' is the y-intercept.
- Point-Slope Form: y - y₁ = m(x - x₁), where 'm' is the slope and (x₁, y₁) is a point on the line.
- Two-Point Form: y - y₁ = [(y₂ - y₁)/(x₂ - x₁)](x - x₁), used when two points (x₁, y₁) and (x₂, y₂) are known.
- Intercept Form: x/a + y/b = 1, where 'a' is the x-intercept and 'b' is the y-intercept.
- General Form: Ax + By + C = 0, where A, B, and C are constants.
- Normal Form: x cos(ω) + y sin(ω) = p, where 'p' is the perpendicular distance from the origin and 'ω' is the angle this perpendicular makes with the positive x-axis.
3. How do you find the equation of a straight line if you are given two points on it?
To find the equation of a straight line from two given points, (x₁, y₁) and (x₂, y₂), you first calculate the slope (m) of the line using the formula: m = (y₂ - y₁) / (x₂ - x₁). Once you have the slope, you can use the point-slope form with either of the two given points. For example, using (x₁, y₁), the equation becomes: y - y₁ = m(x - x₁). Substituting the calculated slope 'm' gives you the final equation.
4. What does the slope of a straight line signify in a real-world example?
In a real-world context, the slope of a straight line represents a rate of change. For example, in a graph plotting distance versus time, the slope signifies the speed of an object. A steeper slope means a higher speed (more distance covered in less time), while a horizontal line (slope = 0) indicates the object is stationary. In a business context plotting profit versus time, the slope would represent the rate of profit growth.
5. How can you determine if two lines are parallel, perpendicular, or intersecting using their equations?
You can determine the relationship between two lines by comparing their slopes (m₁ and m₂):
- Parallel Lines: Two lines are parallel if and only if their slopes are equal (m₁ = m₂). They must also have different y-intercepts.
- Perpendicular Lines: Two lines are perpendicular if and only if the product of their slopes is -1 (m₁ * m₂ = -1). This means one slope is the negative reciprocal of the other.
- Intersecting Lines: If the slopes are not equal (m₁ ≠ m₂), the lines will intersect at a single point. If the product of their slopes is not -1, they intersect but are not perpendicular.
6. Why is the general form Ax + By + C = 0 considered 'general' and useful for all straight lines?
The form Ax + By + C = 0 is considered general because it can represent any straight line on a Cartesian plane, including vertical lines. Other forms, like the slope-intercept form (y = mx + c), fail for vertical lines because their slope 'm' is undefined. In the general form, a vertical line (e.g., x = k) can be written as 1x + 0y - k = 0, which is a valid representation. This universality makes it a fundamental equation in coordinate geometry.
7. What is the method to convert the general equation of a line into the slope-intercept form?
To convert the general equation Ax + By + C = 0 into the slope-intercept form (y = mx + c), you must isolate 'y' on one side of the equation. The steps are:
1. Move the Ax and C terms to the right side: By = -Ax - C.
2. Divide the entire equation by B (assuming B is not zero): y = (-A/B)x - (C/B).
Now, the equation is in the form y = mx + c, where the slope m = -A/B and the y-intercept c = -C/B.
