The slope-intercept form is a linear equation used in algebra and analytic geometry to represent the equation of a straight line. It is especially useful for finding the equation of a line when we know the numerical values of the slope and y-intercept. An equation of a line is one in which all points on that line satisfy the equation. The equation of the line can also be expressed in three other forms.
Point-Slope Form Intersection Form Two-Point Forms
The choice of form depends on the given information and the desired result. However, the slope-intercept form is a popular choice among students because of its simplicity.
We will discuss the definition and type of slope-intercept form in this article and learn how to derive the equation with several examples to understand this concept clearly.
Definition and explanation of slope intercept form
Slope-intercept form is a way to find the equation of a straight line in two dimensions. We will need a slope or slope angle and y-intercept to find the equation of the line through this method. This form is popular because it makes it easy to read the slope and y-intercept without drawing the line on a graph.
Let's break the slope-intercept concept into two parts.
1. The slope of a Line is defined as the ratio of the change in the y-axis to the change in the x-axis. It is represented by the symbol m.
2. The Y-intercept is a point on the y-coordinate where a line intersects the y-axis. The y-intercept coordinates are expressed as (0, b).
Formula for slope-intercept of the line
The following equation of a slope-intercept form can be used to evaluate an equation of a straight line.
y = mx + b
Here:
m = slope or steep line. b = y-intercept (point where a line crosses or cuts the y-axis). (x, y) = The coordinates of a point lie on a line.
Derivation of slope-intercept equation
Let's learn how to get the equation of slope intercept form (y = mx + b).
Consider a line L of slope m that crosses the y-axis at a distance of b units from the origin O.
To find the slope of the line L passing through the points (0, b) and (x, y), we will use the slope formula:
∴ m = (y2 – y1) / (x2 – x1)
Plug the given values into the slope equation.
m = (y – b) / (x – 0)
m = (y – b) / (x)
Multiply 'x' on both sides.
mx = y – b
Add 'b' to both sides of the equation.
mx + b = y – b + b
mx + b = y
(the)
y = mx + b
This is the standard form of the slope section.
Converting the general equation of a straight line to slope-intercept form
The slope-intercept equation can be derived from the general equation of a line in the following way.
The general equation for a straight line is
Ax + By + C = 0
Isolate the value of 'y' on one side of the equation.
With = – Ax – C
Divide 'B' by each term.
y = (- A/B) x + (- C/B)
This is the slope-intercept form.
Here (- A/B) is the slope and (- C/B) is the y-intercept (b) of the line.
Solved Examples Line slope intercept form
Let's figure out how to calculate the equation of a straight line from its slope-intercept form.
Example 1:
Find the equation of a line with slope – 4 and y-intercept 2.
Solution:
Step 1: Determine the slope m and y-intercept b of the line.
Given:
Slope (m) = – 4
Y-intercept (b) = 2
Step 2: Use the slope intercept form and plug the given values into it.
y = – 4x + 2
This is the required equation of a line.
Example 2:
Write the equation of a line passing through the point (-2, 5) with a slope of -2.
Step 1: Determine the slope (m) and y-intercept (b) of the line.
Here,
m = -2
(x, y) = (-2, 5)
Step 2: Plug the given values into the slope-intercept equation to determine b.
5 = -2 (-2) + b
5 = 4 + b
b = 1
Step 3: Now put the value of b and m into the slope intercept form formula.
y = – 2x + 1
This is the equation of the line through the point (-2, 5) with a slope of -2.
Example 3:
Determine the equation of a line passing through the points (2, 4) and (6, 8).
Solution:
Step 1: First calculate the slope of the given line.
m = (8 – 4) / (6 – 2)
m = 4 / 4
m = 1
Step 2: Let's pick the first given point (2, 4) to find the value of b.
∴ y = mx + b
4 = (1) (2) + b
b = 2
Step 3: Plug the value of b and m into the slope-intercept equation.
y = 1x + 2
This is the equation of the line passing through the points (2, 4) and (6, 8).
You can also use a slope intercept calculator to solve the problems of finding the equation of the line to get the results in seconds.
conclusion
In this article, we have discussed the introduction of slope-intercept form and then extended this concept to an advanced level. We explored the formula for the slope-intercept form and learned how to obtain it. We converted the general equation of a line into slope-intercept form. We have solved several examples to help you easily determine the equation of the line.