The slope-intercept form is a way to express the equation of a straight line in two-dimensional Cartesian coordinates. Additionally, it can be used to solve problems involving linear relationships, such as finding the equation of a line that passes through two given points, or determining the point where two lines intersect.
In this article, we will discuss the definition of slope intercept form, Straight line of slope intercept form, Derivation of slope intercept form and also explain the topic with example.
Table of Contents
Definition of Slope Intercept Form
The slope-intercept form is a way of representing a linear equation in two variables, typically written as y = mx + b, hare m is the slope of the line and b is the y-intercept. The slope represents the rate at which y changes with respect to x, while the y-intercept is the point where the line crosses the y-axis.
What is slope intercept form of a straight line?
The slope intercept form is a method for determining a straight line’s equation in the coordinate plane. The equation of a straight line is the relationship that:
- The line’s coordinates must fulfil at every point
- The coordinates of any point that is not on the line will not satisfy
This equation’s solution is straightforward. To find the slope intercept form of a straight line, we need to know the slope, or angle of inclination, of the straight line from the x-axis, as well as the intercept it makes with the y-axis.
Consider a line L with slope m that crosses the y-axis c units away from the center.
In this case, the distance c is referred to as the y-intercept of the given line L.
As a result, the coordinate of a point where the line L intersects the y-axis is (0, c). That is, line L has a slope of m and passes through a fixed point (0, c).
We already know that the equation of a line in point slope form, where (x1, y1) is the point and m is the slope, is:
(y – y1) = m (x – x1)
In this case, (x1, y1) = (0, c)
Now put value of Y1=C and X1=0 in given above equation
(y1-c) =m(x-0)
y1-c=mx
y1= mx+ c
Slope intercept Formula
Formula is given by
Vertical coordinate y = mx + b y-intercept
Horizontal coordinate
Slope of the line
Where:
- y is the dependent variable or output variable
- x is the independent variable or input variable
- m is the slope or gradient of the line, which describes the rate of change between y and x.
- b is the y-intercept, which is the value of y when x = 0.
It is called “slope-intercept” because the slope (m) and y-intercept (b) of the line are easily identified and can be used to graph the line.
The slope-intercept formula is widely used in algebra and geometry and is used to model and analyze relationships between variables in many different fields, such as physics, engineering, economics, and more.
Using a slope-intercept form, how do you find the solution for a straight line?
The slope-intercept form of a straight line’s equation is y = mx + b, where m is the slope of the line and b is the y-intercept.
To calculate the equation of a straight line using the slope-intercept form, follow these steps:
- Determine the slope of the line. The slope of a line can be calculated by finding the change in y divided by the change in x between two points on the line. The slope formula is:
m = (y2 – y1) / (x2 – x1)
- Determine the y-intercept. To find the y-intercept, you can either use a given point on the line or the slope and another point on the line.
- Write the equation. Once you have determined the slope and the y-intercept, write the equation in slope-intercept form by substituting the values of m and b into the equation y = mx + b.
For example, if the slope of a line is 2 and the y-intercept is (0, 3), the equation in slope-intercept form is y = 2x + 3.
Example Section
Example 1:
Find the equation of the straight line that has slope m = 5 and passes through the point (–3, –9).
Solution:
By the slope-intercept form we know;
y = mx +c
Step 1:
Given,
m = 5
As per the given point, we have;
y = -9 and x = -3
Step 2:
Hence, putting the values in the above equation, we get;
-9 = 5(-3) + c
-9 = -15+c
c = -9 + 15= 6
Hence, the required equation will be;
y = 5x+6
You can also take assistance from a y=mx+b calculator to find line equation through slope intercept form with steps.
Example 2:
Find the equation of the lines for which sin θ = 1/3, where θ is the inclination of the line such that:
- y-intercept is -7
- x-intercept is 5/3
Solution:
We know that the inclination of a line with slope m is given by θ = tan⁻¹(m). If Tan θ = 1/3, then we can write:
m = tan θ = 1/3
- the y-intercept is -7:
If the y-intercept of the line is -7, then the line passes through the point (0, -7).
Step 1:
So the equation of the line is:
y = mx + b, where m = 1/3 and b = -7
Step 2:
Substituting the values, we get:
y = (1/3) x – 7
This is the equation of the line with inclination θ such that Tan θ = 1/3 and y-intercept -7.
- the x-intercept is 5/3:
If the x-intercept of the line is 5/3, then the line passes through the point (5/3, 0).
Step 1:
So the equation of the line is:
y – y1 = m (x – x1),
where m = 1/3 and (x1, y1) = (5/3, 0)
Substituting the values, we get:
y – 0 = (1/3) (x – 5/3)
Step 2:
Simplifying, we get:
y = (1/3) x – 5/9
This is the equation of the line with inclination θ such that Tan θ = 1/3 and x-intercept 5/3.
Conclusion
In this article, we have discussed the definition of slope intercept form, Straight line slope intercept form, using slope intercept how you find the solution of the straight line, and also with the help of an example topic will be explained. After studying this article everyone can defend this topic.