Slope Intercept Form - Formula, Derivation, and Calculation

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The slope-intercept form is used to express the equation of a straight line. We can use different methods to represent the equation of the straight line, but it depends on the given information. The slope-intercept form uses when we have the slope and y-intercept of the line.

The equation of the straight line can be expressed in different forms, and each form is useful in different ways. Some common methods are here to determine the equation of a line:

• Point slope form
• Two-point form
• Slope intercept form
• Intercept form

In this article, we will confine ourselves to slope-intercept form. We will discuss the definition of intercept form and its formula. We will learn how to originate its formula. We will solve some examples related to the slope-intercept form.

How to define a slope-intercept form?

A slope-intercept form is used to write a straight-line equation by using slope and y-intercept (the point at which it cut the y-axis). Mathematically, it is stated as y = mx + b, where m is the slope and “b” is the y-intercept of a line.

The slope “m” is defined as the ratio of the net change on the y-axis to the net change on the x-axis. The y-intercept “b” is the point where the line cut the y-axis. This means finding the number for y when x is zero. The y-intercept provides us with a starting point on the y-axis and helps us determine the line’s position vertically.

The equation of a straight line can be easily written by using the slope and y-intercept of the line.

The equation of the Slope-intercept form

The formula for the slope-intercept form to find the equation of a straight line is given below;

y = mx + b

Here are;

• y represents the Dependent variable, usually plotted on the vertical axis
• x represents the Independent variable, usually plotted on the horizontal axis
• m = Slope or steepness of the line
• b = It is the point where the line cut the y-axis. (Where x = 0)

How to derive the equation of the slope-intercept form?

If the line passes through the point (x1, y1) and (x, y) is any other point on the line, then slope “m” is defined as the ratio of the net change on the y-axis to the net change on the x-axis. So, the definition of slope

m = (y – y1) / (x – x1) ______ (1)
 

Let’s learn how can we derive the equation of the slope-intercept form. Consider a straight line L having slope m cut the y-axis at a distance of b units from the origin. 

 

 

As, (x1, y1) = (0, b)

Substitute these values in 1. We get,

m = (y – b) / (x – 0)

m = (y – b) / x

After, x Multiplying both sides,

mx = (y – b)

After, rearranging it, we have

y = mx + b _____ (A)

Equation “A” is the equation of slope intercept form.

Slope-intercept equation from the standard form of the linear equation

We know that the standard form of linear equation is

Ax + By = C _____ (2)

Where A, B, and C are integers additionally A is a non-negative integer.

After rearranging 2 to get

By = – Ax + C

Divide “B” by both sides

y = (–A/B) x + (C/B)

Here, (–A/B), and (C/B) represent the slope, and y-intercept of the line respectively.

Solved Examples of Slope-intercept form

In this section, we will learn how to find the equation of a straight line by using the slope-intercept form.

Example 1:

Determine the equation of a straight line, when the y-intercept Is 3 and the slope is 4.

Solution: 

Here;

Y-intercept = b = 3

Slope = m = 4

To obtain the equation of the straight line, put the given values in the formula of slope intercept form. I.e.

y = 4x + 3

That is the equation of a straight line.

Example 2:

If a line passes through a point (2, 4) with slope 5. Then calculate the equation of a straight line.

Solution: 

In this example, we have

Slope = m = 5

x = 2

y = 4

Substitute these values in the equation of slope intercept form

∴ y = mx + b

4 = (5) (2) + b

4 = 10 + b

b = – 6

Now, put b = – 6, and m = 5 in the equation of slope intercept form.

y = 5x + (– 6)

y = 5x – 6

That is our required equation.

You can also take assistance from the online y=mx+b calculator to find slope intercept form with the help of the slope and point of the line.

Example 3:

If a straight line passes through a point (2, 4) and (6, 8). Then evaluate the equation of a straight line.

Solution: 

Here,

x = 2 & x1 = 6

y = 4 & y1 = 8

First, we will find the slope of the given line.

∴ m = (y – y1) / (x – x1)

m = (4 – 8) / (2 – 6)

m = – 4 / – 4

m = 1

Let’s choose the first given point (2, 4) and find the value of b.

∴ y = mx + b

4 = (1) (2) + b

4 = 2 + b

b = 2

Now, put the value of b and m in the equation of slope intercept form.

y = 1x + 2

That is a straight line that passes through points (2, 4) and (6, 8).

Conclusion

In this article, we have discussed the slope intercept form, which is the common method used to determine the equation of a straight line. We discussed its formula and described its entire notation. We learned how to derive the slope intercept form by using the definition of slope and the standard form of linear equation.

We did solve some examples of slope intercept form in which we determine the equation of the straight line by using the equation of slope intercept form. After reading this article, you will be able to find the equation of a straight line by using the slope intercept form.

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