Methods to write the equation of a line?

The equation of a line is the equation that is satisfied by each point that lies on that line. There are various methods to find this equation of a straight line given as,

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

What is the slope-intercept form?

To write the equation of a line there are so many ways but the most common way is called the slope-intercept form.  It’s called slope-intercept form because it clearly shows the slope and the y-intercept in the equation. The slope is the value written with x mean coefficient of x is a slope and the constant written at the end is called y- the intercept.

Slope

A slope is basically a number that measures the steepness and direction of the line and it is a ratio of the vertical change of the graph to the horizontal change. Its formula is rise/ run and reads as “rise over run”.  As an illustration, if the slope of a line is 2/3, that means it goes up 2 units for every 3 units that it goes over.

Intercept

Basically, there are two axes on the coordinate plane which are the x and y-axis.  When a line crosses one of the given axes then the point where it crosses is called an intercept. Coming towards the slope-intercept form, the equation of the line uses the y-intercept for its working where the line crosses the y-axis. On the other hand, the y-axis is the vertical axis that goes up and down.  The y-intercept is the point where the line and y-axis touch.

When it comes to graphing lines on the coordinate plane, there are a few different ways to show a linear equation and check out how to find a given line’s slope, y-intercept, points it passes through, and how its graph looks like. For better understanding and solving problems involving linear equations in various forms, including point-slope form, these are necessarily useful, and important algebra skills that will help you solve math problems involving linear equations. 

The discussion below will give you a detailed explanation step-by-step through solving point-slope formula examples (sample problems) that we will solve and find the final correct answer together. By working through the help of examples, you will gain a much better understanding of the point-slope formula, its purpose, and methods to use it to solve problems on upcoming work assignments, quizzes, tests, and exams.

Derivation of Slope-Intercept Form by using Standard Form Equation

You can make 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 for a straight line is:

P(x) + Q(y) + R = 0 where P, Q and R the constants 

Re-arranging the terms as:

Q(y) = -P(x) – R

⇒y = (-P/Q) (x) + (-R/Q)

This is of the form y = m(x) + c

Here, (-P/Q) represents the slope of the line, and (-R/Q) is the y-intercept.

Formula

Using the above formula of slope-intercept, the given equation of the line is:

y = mx + b

Here,

• m = for showing the slope of the line
• b = for showing the y-intercept of the line
• x, y shows every point on the line
• x and y have to be used as the variables while applying the above formula of slope intercept.

What are the Applications of the Slope-Intercept Formula?

The slope-intercept formula is basically used for the following things given below:

• To determine the equation of a line.
• To make a graph of a line using the y-intercept and slope.
• To calculate the slope of a line easily.
• To calculate the intercepts of a line easily.

What is point-slope form?

Point slope form is commonly used to show a straight line using its slope value “m” and given point on the line. Many methods can be applied to express the equation of a straight line and it is one of them.

Definition

“The point-slope form of a line is expressed using the slope of the line and point that the line passes through”

Derivation of the Point slope formula 

Let’s have a look at how to find the point-slope form (i.e. the proof of the formula of the point-slope form). We will derive the point-slope formula using the equation for the slope of a line. Let us take a line whose slope is “m”. Suppose we assume that (x₁, y₁) is a given point on the line. Suppose (x, y) be any other arbitrary point on the line whose coordinates are unknown.

The way to find the equation for the slope of a line is 

Slope = (change in y-coordinates) / (change in x-coordinates)

m = (y – y1)/(x – x1)

Multiply both sides by (x – x1) it will get

m(x – x1) = y – y1

It also can be written as,

y – y1 = m(x – x1)

Thus the point-slope formula is derived.

Formula:

y- y1 = m(x – x₁)

Here m is the slope of the line and x1 and y1 are equal to the corresponding x- and y-coordinates of a given point the line passes through.

Unless slope-intercept forms, point-slope form does not demand you to know the function’s y-intercept. In spite of this, you only deal with the slope and the coordinates of a single point that the line passes through.

If you find uneasiness to understand these definitions analyze the key similarities and differences between the two forms.

Method to calculate Point slope form?

To find a point-slope form for a given straight line for determining the equation of the given line, we can do the steps written below,

Step 1: write the slope, ‘m’ of the straight line, and the coordinates (x1, y1) of the required point that lies on the line.

Step 2: Substitute all values in the point-slope formula y – y1= m(x – x1).

Step 3: Express them to obtain the equation of the line in standard form.

Example Sections

In this section with the help of examples, the equation of straight lines will calculate with the help of point slope-intercept form.

Example 1: 

Write the equation of the straight line that has slope m = 4 and passes through the point (1, 6).

Solution:

As we know the equation of Slope-Intercept form 

y = m(x) +c

Step 1: Given data

Slope = m = 4

As per the given point, we have the following;

y = 6 and x = 1

Step 2: Putting the values in the slope-intercept form you will get

6 = 3(1) + c

6 = 3+c

c = 6-3 = 3

Hence, the given equation of the straight line would be;

y = 4x+3

Example 2:

Write the equation of the straight line that has slope m = -2 and passes through the point (4, 2).

Solution:

As we know the equation of slope intercepts form

y = m(x) +c

Step 1: Given,

m = -2

As per the given point, we have the following;

y = 4 and x = 2

Step 2: Putting the values in the slope intercepts form you will get

4 = -2(2) + c

4 = -4 + c

c = 4+4 = 8

Hence, the required equation of a straight line would be;

y = -2x+8

Example 3:

Write the equation of the lines for which tan θ = 1/3, where θ is the inclination of the line such that: 

(i) y-intercept is 5

(ii) x-intercept is 1/3

Solution:

Step 1: Extract the given data here first

Slope = tan θ = 0.5

 (i) y-intercept = c = 5

Step 2: As you know using slope intercept forms an equation of the line 

y = m(x) + c

y = (1/3) (x) + (5)

Or

3y = x +15

x – 3y +15 = 0

Step 3: 

(ii) x-intercept = d = 1/3

As you know using slope intercept forms an equation of the line 

y = m(x – d)

y = (1/3) [x – (1/3)]

Or

3y = (3x – 1)/3

9y = 3x – 1

3(x) – 9y – 1 = 0

Example 4:

Write 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 6 

(ii) x-intercept is 1/3

Solution:

Step 1: Extract the given data here first

Slope = tan θ = 0.5

 (i) y-intercept = c = 6

Step 2: As you know using slope intercept forms an equation of the line 

y = m(x) + c

y = (1/2) (x) + (6)

Or

2y = x +12

x – 2y +12 = 0

Step 3:

 (ii) x-intercept = d = 1/3

As you know using slope intercept forms an equation of the line 

y = m(x – d)

y = (1/2) [x – (1/3)]

Or

2y = (3x – 1)/3

6y = 3x – 1

3(x) – 6y – 1 = 0

Example 5: 

Calculate the equation of a line that passes through a point (4, -6) and has a slope is (-1/2).

Solution:

Step 1: Extract the given data here first

The point on the given line is (x1, y1) = (4, -6)

Slope = m = (-1/2)

Step 2: The equation of the line is given using the point-slope form:

y − y1 = m(x − x1)

Step 3: Putting the values of slope and given points in the point-slope form

y − (−6) = (−1/2)(x − 4)

y + 6 = (−1/2) (x) + 2

Minus 6 from both sides,

y = (−1/2) (x) – 4

y = −0.5x − 4

0.5x + y + 4 = 0

Hence, the equation of the given line is, 0.5x + y + 4 = 0

The point slope form calculator by Allmath [https://www.allmath.com/pointslopeform.php] can also be used to write the equation of the line through points and the slope of the line to avoid calculations.

Conclusion

For showing linear equations, the point-slope form can be considered the fastest. By knowing the slope and a single point that lies on a straight line you can easily find the point-slope of an equation. The information which is gathered by a straight line may change and you can utilize the necessary formulas for changing graphs to show what you want.