In geometry, the equation of a line can be written in different forms and each of these representations is useful in different ways. The equation of a straight line is written in either of the following methods:
In this article, you will learn about one of the most common forms of the equation of lines called slope-intercept form along with derivation, graph and examples.
Learn what is the intercept of a line here.
Let’s have a look at the slope-intercept form definition.
The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form , and the values of m and c are real numbers.
The slope, m , represents the steepness of a line. The slope of the line is also termed as gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.
In this section, you will learn the derivation of the equation of a line in the slope-intercept form.
Consider a line L with slope m cuts the y-axis at a distance of c units from the origin.
Here, the distance c is called the y-intercept of the given line L.
So, the coordinate of a point where the line L meets the y-axis will be (0, c).
That means, line L passes through a fixed point (0, c) with slope m.
We know that, the equation of a line in point slope form, where (x1, y1) is the point and slope m is:
Substituting these values, we get;
Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c
Note: The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively.
As derived above, the equation of the line in slope-intercept form is given by:
(x, y) = Every point on the line
m = Slope of the line
c = y-intercept of the line
Usually, x and y have to be kept as the variables while using the above formula.
We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:
m = Slope of the line
d = x-intercept of the line
Sometimes, the slope of a line may be expressed in terms of tangent angle such as:
We can derive the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below:
As we know, the standard form of the equation of a straight line is:
Rearranging the terms as:
This is of the form y = mx + c
Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.
When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form. Since it is in the form “y=”, hence it is easy to graph it or solve word problems based on it. We just have to put the x-values and the equation is solved for y.
The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.
Example 1:
Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5).
Solution:
By the slope-intercept form we know;
As per the given point, we have;
Hence, putting the values in the above equation, we get;
Hence, the required equation will be;
Example 2:
Find the equation of the straight line that has slope m = -1 and passes through the point (2, -3).
Solution:
By the slope-intercept form we know;
As per the given point, we have;
Hence, putting the values in the above equation, we get;
Hence, the required equation will be;
Example 3:
Find the equation of the lines for which tan θ = 1/2, where θ is the inclination of the line such that:
(i) y-intercept is -5
(ii) x-intercept is 7/3
Solution:
Given, tan θ = 1/2
So, slope = m = tan θ = 1/2
(i) y-intercept = c = -5
Equation of the line using slope intercept form is:
(ii) x-intercept = d = 7/3
Equation of slope intercept form with x-intercept is:
Quiz on Slope Intercept Form
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