A straight line’s slope intercept form is one of the most frequent ways to represent the equation of a line. When given the slope of a straight line and the y-intercept, the slope intercept formula can be used to derive the equation of a line ( the y-coordinate of the point where the line intersects the y-axis). However, the equation of a line is the equation that each point on that line must satisfy.
Moreover, there are several ways to find this equation for a straight line, which is written as,
- Slope-intercept form
- Also, Point slope form
- Then, Two-point form
- Then, Intercept form
However, let’s look at how to calculate the slope intercept formula and how to deduce it using examples.
What is the Slope Intercept Form of a Straight Line?
A method for determining the equation of a straight line in the coordinate plane is the slope intercept form. The equation of a straight line is also the connection that any point on the line’s coordinates must satisfy.Read Also: What is the derivative of sin^2x ?
However, any point not on the line’s coordinates will not satisfy.
However, it’s simple to figure out how to solve this equation. We’ll need the slope, or angle of inclination, of this straight line from the x-axis, as well as the intercept it makes with the y-axis, to get the slope intercept form of a straight line.
Slope Intercept Form Definition
However, to find a line’s equation, use the slope-intercept form. The slope of the line and the intercept cut by the line with the y-axis are required for the slope-intercept formula to work. However, consider the slope ‘m’ and the y-intercept ‘c’ of a straight line. Moreover, for a straight line with a slope of ‘m’ and a y-intercept of ‘c,’ the slope intercept form equation is y = mx + c.
Forms of the Slope Intercept
The slope intercept form is illustrated with a few instances.
However, the equation for a line with a slope of -1 and a y-intercept of (4) is y = -x + 4.
However, y = 2x is the equation for a line with slope (2) and passing through the origin (y-intercept = 0).
Note: Tan can be used to compute the slope of the line for which the angle of inclination is given. In addition, if we have two points on a straight line (x1, y1) and (x2, y2), the slope can be written as (y2 – y1)/ (x2 – x1). However, for a deeper grasp of the notion, consider the slope-intercept formula and its derivation.
Slope Intercept Formula
Given the required parameters, the slope-intercept formula is also used to compute the slope, y-intercept, x-intercept, or equation of a straight line. However, the equation of a straight line can be found using a variety of formulas. When we know the slope of a straight line, which is denoted by m, and the y-intercept of the straight line, which is denoted by c or d, we use the slope-intercept formula (0, c). However, let’s look at a few examples of the slope-intercept formula. The slope-intercept formula is shown below.
Slope Intercept Form Calculator
If you know two places that a line passes through, the slope intercept form calculator can assist you in determining its equation. The slope intercept form calculator will show you how to find a line’s equation from any two locations through which it passes. Using the slope intercept formulas, you may find the slope coefficients and y-intercept, as well as the x-intercept. Continue reading to understand what the slope intercept form of a linear equation is, how to find the equation of a line, and why the slope intercept form equation is important in real life.
Slope Intercept Formula in Math
Using the slope-intercept formula, the equation of the line is:
y = mx + c
where,
m = the slope of the line
Then, c = y-intercept of the line
Every point on the line is represented by (x, y). When using the given formula, the variables x and y must be kept in mind.
However, it is important to note that the slope-intercept formula cannot be used to determine the equation of a vertical line. Here’s an illustration of how to use the slope intercept formula.
Consider the following scenario:
A line’s equation is 3x + 4y + 5 = 0. However, using the slope intercept form, find the line’s slope and y-intercept.
Solution:
We rewrite the line’s equation in the usual form y = mx + c via rearrangement.
We have:
4y = -3x – 5
Then, y = (-3/4)x + (-5/4)
Thus, m = -3/4 , c = -5/4
Answer: The slope of the given straight line, m = -3/4 and the y-intercept, c = -5/4.
Slope Intercept Form of a Line
To get the equation of a line with an arbitrary inclination, we’ll need two things: the line’s inclination (or its slope or the angle it makes with, say, the x-axis) and the line’s placement (i.e. where it passes through with respect to the axes). By specifying the point on the y-axis through which the line passes, or in other words, by specifying the y-intercept, c), we can describe the line’s location. These two characteristics can be used to determine the uniqueness of any line.
The steps to finding a line’s equation using the slope-intercept form are as follows:
Step 1:
Write down the y-intercept, ‘c,’ and the line’s slope,’m.’ If the slope of a straight line is not given explicitly and other necessary data is provided, we can use the slope formula to find it.
Step 2:
However, write y = mx + c in the slope intercept formula.
Example: A 60° angle to the horizontal is formed by a line passing through the point (0, – 1). Hence, determine the line’s equation.
Solution: We obtain m = tan 60° = √3 as our result.
As a result, the line’s equation is y = mx + c.
⇒y = (√3)x + (−1)
⇒y = √3x − 1
Standard Form to Slope Intercept Form
However, by rearranging and comparing, we can change a line’s equation from standard form to slope intercept form. We know that Ax + By + C = 0 is the conventional form of the equation of a straight line. Likewise, we get B y = -Ax – C by rearranging the terms to find the value of ‘y’.
Then, y = (-A/B)x + (-C/B),
Then, the slope of the line is (-A/B), and the y-intercept is (-C/B).
Slope Intercept Form Equation
In this section, you’ll learn how to deduce the slope-intercept form of a line’s equation.
However, consider a line L with a slope of m that intersects the y-axis at c units from the origin.
The y-intercept of the given line L is defined as the distance c.
As a result, the coordinates of a point on the y-axis where the line L intersects will be (0, c).
That is, line L has a slope of m and passes through a fixed point (k0, c).
However, the equation of a line in point slope form, where the point is (x1, y1) and the slope is m, is:
(y – y1) = m(x – x1)
Then, here, (x1, y1) = (0, c)
Moreover, substituting these values, we get;
y – c = m(x – 0)
Then, y – c = mx
Similarly, y = mx + c
As a result, if and only if y = mx + c, the point (x, y) on the line with slope m and y-intercept c lies on the line.
Note that depending on whether the intercept is put on the positive or negative side of the y-axis, the value of c might be positive or negative.
Slope Intercept Form Formula Symbols
The equation of the line in slope-intercept form, as derived above, is:
y = mx + c
Here,
(x, y) = Every point on the line
Then, m = Slope of the line
Then, c = y-intercept of the line
When utilising the aforementioned formula, the variables x and y must usually be preserved as variables.
The coefficients in slope-intercept form
However, aside from being tidy and simple, the slope-intercept form has the advantage of revealing two key characteristics of the line it represents:
- The slope is ‘m’.
- Then, the y coordinate of the y intercept is c. In other words, the line’s y intercept is at (0, c).
However, the line y= 2x + 1 has a slope of 2 and a y intercept of 1 as an example (0, 1)
The fact that this form provides both the slope and the y intercept is why it is referred to as slope-intercept in the first place.
Slope Intercept Form x Intercept
The slope-intercept version of the equation for line L with slope m and x-intercept d can be written as:
y = m(x – d)
Here,
m = Slope of the line
Then, d = x-intercept of the line
Hence, the slope of a line is sometimes stated in terms of tangent angle, as in:
m = tan θ
Derivation of Slope Intercept Form from Standard Form Equation
However, from the equation of a straight line in standard form, we may get the slope-intercept form of the line equation:
Similarly, as we all know, the conventional form of a straight line equation is:
Ax + By + C = 0
Rearranging the terms as:
By = -Ax – C
Then, y = (-A/B)x + (-C/B)
However, this is of the form y = mx + c
Likewise, here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.
Slope Intercept Form Graph
We get a straight line when we draw the graph for the slope-intercept form equation. The optimum form is slope-intercept. It is simple to graph or solve word problems based on it because it is in the form “y=”. All we have to do now is plug in the x-values, and the equation for y is solved.
The best thing about the slope-intercept form is that we can pull the slope and intercept values right out of the equation.
Slope Intercept Form Calculator with Two Points
- Calculate the slope-intercept form equation of the line passing through (0, 1) and (3, 5).
Step 1: Calculate the slope (m).
Remember that the slope is calculated by dividing the change in y by the change in x. (rise over run).
M = (Y2 – Y1) / (X2 – X1)
Slope Intercept Form Calculator with One Point
We can skip right to calculating the y-value of the y-intercept when asked to formulate a linear equation in slope-intercept form (y=mx+c) using the slope and one point on the line.
We may return to the generic slope-intercept form formula and substitute m and c with their values once we’ve determined the slope and y-value of the y-intercept.
Let’s look at some instances of how to find the slope-intercept form of a line given a slope and a single point.
EXAMPLE NO. 1
To begin, use this slope and point to determine the slope-intercept form:
Slope: 3
Point: (2, -6)
Step 1. Solve for the y-value of the y-intercept (c).
Plug the given point’s slope, x, and y into y = mx + c, then solve for c.
Slope: 3
Point: (2, -6)
Y= mx + c
-6 = 3×2 + c
Or, c = -12
Slope Intercept Form Examples
Example 1:
Find the equation of the straight line that goes through the point (–2, –5) and has a slope of m = 3.
Solution:
By the slope-intercept form we know;
y = mx+c
Given,
m = 3
As per the given point, we have;
y = -5 and x = -2
Hence, putting the values in the above equation, we get;
-5 = 3(-2) + c
-5 = -6+c
c = -5 + 6 = 1
Hence, the required equation will be;
y = 3x+1
Example 2:
Find the equation of the straight line that passes through the point and has a slope of m = -1. (2, -3).
Solution:
By the slope-intercept form we know;
y = mx+c
Given,
m = -1
As per the given point, we have;
y = -3 and x = 2
Hence, putting the values in the above equation, we get;
-3 = -1(2) + c
-3 = -2 + c
c = -3+2 = -1
Hence, the required equation will be;
y = -x-1
Example 3:
Find the equation of the lines for which tan = 1/2, where is the line’s inclination and:
(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:
y = mx + c
y = (1/2)x + (-5)
Or,
2y = x – 10
x – 2y – 10 = 0
(ii) x-intercept = d = 7/3
Equation of slope intercept form with x-intercept is:
y = m(x – d)
y = (1/2)[x – (7/3)]
Or,
2y = (3x – 7)/3
6y = 3x – 7
3x – 6y – 7 = 0
Example 4:
Find the equation of a straight line whose slope is 1/3 and whose y-intercept is using the slope intercept form (0, -5).
Solution:
To obtain the equation of the given line, do the following:
Given: the line has a slope of m = 1/3.
The line’s y-intercept is (0, b) = (0, -5) b = -5.
The equation of the given line is, using the slope-intercept formula.
y = mx + b
y = (1/3) x – 5
Answer: The equation of the given line is, y = (1/3) x – 5.
Example 5:
Find the equation for the horizontal line that crosses the y-axis at the point (0, 3). Use the slope-intercept formula to solve it.
Solution:
To obtain the equation of the given line, do the following:
It is assumed that the line’s y-intercept is (0, b) = (0, 3) b = 3.
The slope of the line is m = 0 because it is horizontal.
The equation of the given line is, using the slope-intercept formula.
y = mx + b
Then, y = 0x + 3
Hence, y = 3
Answer: The equation of the given line is, y = 3.
Example 6:
Find the equation for a line that is parallel to y = 3x – 5 and has a (-1/5) y-intercept.
Solution:
To locate: The equation of a line that is perpendicular to the provided line.
We provide the y-intercept of the line as B = -1/5.
The provided line’s equation is y = 3x – 5.
When we compare this to y = mx + b, we find that its slope is m = 3.
The slopes of the given and necessary lines are equal since they are parallel.
As a result, M = 3 is also the slope of the needed line.
Using the slope-intercept formula, the needed line’s equation is,
y = Mx + B
y = 3x – 1/5
Answer: The equation of the required line is, y = 3x – 1/5.
FAQs
What is Slope Intercept Form in Math?
In mathematics, the slope intercept form is one of the methods for calculating the equation of a straight line. Slope and the y-axis intercept of the equation is the only requirement in that case. However, y = mx + b is the slope intercept form, where ‘m’ is the straight line slope and ‘b’ is the y-intercept.
How do you find slope intercept form?
The slope intercept form, in general, yields the formula: y = mx + c.
We represent the slope by M. (lesson on the slope). The letter ‘m’ stands for ‘move.’
Similarly, we can denote the y-intercept as the letter c. (lesson on the y-intercept). However, the letter ‘c’ stands for the beginning of the line.
What is the Slope Intercept Form Equation?
Using a straight line’s slope and the point where it intersects the y-axis, we can use the slope intercept equation to obtain the general equation of the line. However, y = mx + b is the slope intercept form equation.
How do you find y MX B?
To find the y-intercept, b, use the formula y = mx + b. Replace the coordinates of one of the supplied points with x and y in the formula, and the calculated value with m. (2). However, find the equation of the line whose graph includes the points (1,–2) and (6,5).
How to Find the Equation of a Straight Line Using Slope Intercept Form?
To find the straight-line equation using the slope intercept form, we require the straight line’s slope and its point of intersection with the y-axis. Similarly, we can use the slope formula to determine the slope of a line. However, the equation of the straight line can be determined using the slope intercept form as y = mx + b, where, m is the straight line’s slope and ‘b’ is the y-intercept.
How do you find the slope of Y MX B?
We can write any straight line’s equation as y = mx + b, where m is the slope and b is the y-intercept. It is also applicable in case of linear equations. The value of y at the point where the line crosses the y axis is the y-intercept of this line.
What is the Slope-Intercept Formula?
One of the formulas used to obtain the equation of a line is the slope-intercept formula. However, y = mx + b is the slope-intercept formula for a line with slope m and y-intercept b. Similarly, (x, y) determines any point on the line.
How do I find slope?
The slope of a line determines the direction of it. However, divide the difference in the y-coordinates of two points on a line by the difference in the x-coordinates of those same two points to get the slope.
What are the Applications of the Slope-Intercept Formula?
We use the slope-intercept formula to determine a line’s equation.
- Using the y-intercept and slope, draw a line.
- Easily determine the slope of a line.
- It’s simple to find the intercepts of a line.
How to Find the Slope of a Line Using the Slope-Intercept Form?
The slope of a line can be calculated using the slope-intercept form, y = mx + b, where ‘m’ is the line’s slope and ‘b’ is the y-intercept. However, here’s an illustration. Let’s calculate the slope of the line 6x 3y = 5. To get into the slope-intercept form, let’s solve this for ‘y.’ Then we get y = 2x – (5/3) as a result. When we compare this to the slope-intercept formula, y = mx + b, we get m = 2.
How to Convert Standard Form of Straight Line Equation to Slope Intercept Form?
Ax + By + C = 0 is the usual form of the equation of a straight line. We may find the slope intercept of any straight line provided in this form: y = (-A/B)x + (-C/B), where (-A/B) represents the line’s slope and (-C/B) represents the y-intercept.