Straight Line Equation

 

Straight Line Equation: Definition, Forms, & Examples

In geometry, the equation of a straight line is a well-known way of representing linear equations. There are various ways to represent the equation of the line such as standard form of line, two points form, two points intercept form, etc.

The general and well-known way of representing a straight-line equation is y = mx + b, where the slope of the line is represented by “m” and the y-intercept is represented by “b”. In this article, we’ll explore the definition, different forms, and examples of straight-line equations.

What is the Straight Line Equation?

In geometry, the straight-line equation is a widely used concept for writing equations of two variable linear equations. It is a linear equation that involves two variables i.e., x&y that satisfies each point on the line.

Usually straight-line equation is a mathematical equation that provides the relationship between the coordinate points of the x-axis and y-axis points that are lying on a line.

There are various forms of representing the equation of the straight line that allow us to get the slope, y-intercept, and x-intercept. The points of the line are also calculated by using the straight-line equation if the other parameters are present.

Formulas to determine the slope and y-intercept of the line.

The slope of the line is calculated by taking the quotient of change in the values of the y-axis and change in the values of the x-axis. It is represented by “m” and mathematically it can be written as:

m = change in y values / change in x values

Where

·         m = slope of the line

_·          change in y values = y₂ – y₁

_·          change in x values = x₂ – x₁

Forms of Straight Line Equation

Usually, the straight-line equation is determined by the standard form, point-slope form, and slope-intercept form. Let’s explore the different forms of straight-line equations.

Point Slope Form

The point slope form is one of the forms for representing a straight-line equation. This form of equation is found by using the slope of the line and a pair of points (x₁, y₁) to represent the line equation. Mathematically, the line equation through point slope form is:

y – y₁ = m (x – x₁)

Where

·         m = slope of the line

_·          (x₁, y₁) = coordinate points of the line

_·          (x, y) = arbitrary point on the line

How to derive the point-slope form Equation?

The equation of point slope form can be derived by using the formula of slope of the line. Let us assume the slope “m” of the line, (x₁, y₁) is the point on the line, and (x, y) be any random points on the line. Take the expression of the slope of the line:

m = change in y values / change in x values

m = (y – y₁) / (x – x₁)

Now multiply the above equation by (x – x₁) to eliminate it from the right.

m (x – x₁) = (y – y₁) (x – x₁) / (x – x₁)

⇒m (x – x₁) = (y – y₁)

The above equation can also be written as:

(y – y₁) = m (x – x₁)

Which is the straight line equation through point slope form.

Slope Intercept form

The slope intercept form is a well-known and general form of representing straight-line equation. In this form, the equation is represented with the help of slope “m” and y-intercept “b”. Mathematically, the line equation through slope intercept form is:

y = mx + b

Where

·         m = slope of the line

_·          b = y-intercept of the line

_·          (x, y) = arbitrary point on the line

How to derive the slope intercept form Equation?

The equation of slope intercept form can be derived by using the formula of slope of the line. Let us assume the slope “m” of the line, (x₁, y₁) = (0, b) is the point on the line, and (x, y) be any random points on the line. Take the expression of the slope of the line:

m = change in y values / change in x values

m = (y – y₁) / (x – x₁)

Now place (x₁, y₁) = (0, b)

m = (y – b) / (x – 0)

⇒m = (y – b)/(x)

Now multiply the above equation by (x) to eliminate it from the right.

m (x) = (y – b)(x) /(x)

m (x) = y – b

The above equation can also be written as:

y = mx + b

Which is the straight-line equation through slope intercept form

Standard Form of Line

The standard form of a line is one of the forms of representing a line equation. In this form, the equation is written in a linear way by using real numbers and variables. Mathematically, the line equation through the standard form of a line is:

Ax + By + C = 0

Where

_·          A, B, & C = real numbers

_·          (x, y) = arbitrary point on the line

Other forms of straight line equation

Below are a few other forms of straight-line equation.

Forms of line equation

	General Equation
Intercept form	x/A + y/B = 1
Two Points Form	y - y1 = [(y2 - y1)]* 1 / [(x - x1) (x2 - x1)]
Normal Form	p = x cos(θ) + y sin(θ)

How to Find the Straight Line Equation?

The equation of the straight line can be determined by various forms. We’ll solve a few examples of each form for a better understanding of the concept.

Examples of Point Slope Form

Example 1: For two coordinate Points

Determine the equation of the line by using the given coordinate points of x & y (12, -5) & (16, 4).

Solution

Step 1:Write the given points of the x-axis and y-axis.

x₁ = 12, x₂ = 16, y₁ = -5, y₂ = 4

Step 2:Now calculate the slope of the line in order to find the straight-line equation.

Slope of the line = m = [y₂ – y₁]/ [x₂ – x₁]

Slope = m = [4 – (-5)] / [16 – 12]

Slope = m = [4 + 5] / [4]

Slope = m = 9/ 4

Slope = m = 2.25

Step 3:Now write the general expression of point slope form and place the value of the slope of the line and a coordinate point on the line to get the straight-line equation.

y – y₁ = m (x – x_1­)

y – (-5) = 2.25 * (x – 12)

y + 5 = 2.25 * (x – 12)

Example 2: For a given slope& point

Determine the point slope form equation if the given slope of the line is -6 which passes through a point (4, -6)

Solution

Step 1:Write the given slope and point of the line.

Slope = m = -6

x₁ = 4

y₁ = -6

Step 2:Now write the general expression of point slope form and place the value of the slope of the line and a coordinate point on the line to get the straight-line equation.

y – y₁ = m (x – x_1­)

y – (-6) = -6 * (x – 4)

y + 6 = -6 * (x – 4)

Examples of Slope Intercept Form

Example 1: For two coordinate points

Determine the equation of the line by using the given coordinate points of x & y

(x₁, y₁) = (2, -1) & (x₂, y₂) = (8, 4)

Solution

Write the given points of the x-axis and y-axis.

x₁ = 2, x₂ = 8, y₁ = -1, y₂ = 4

Step 2:Now calculate the slope of the line in order to find the straight-line equation.

Slope of the line = m = [y₂ – y₁] / [x₂ – x₁]

Put the given values

Slope = m = [4 – (-1)] / [8 – 2]

Slope = m = [4 + 1] / [8 – 2]

Slope = m = [5] / [6]

Slope = m = 0.833

Step 3:Now put one point of the line and the slope of the line in the general expression of slope intercept form in order to find the y-intercept of the line.

y = mx + b

-1 = 5/6 (2) + b

-1 = 1.667 + b

- 1 – 1.667 = b

-2.667 = b

Step 4:Now place the slope of the line and y-intercept of the line to the general expression of slope intercept form to get the straight-line equation. 

y = mx + b

 y = 0.833x + (-2.667)

y = 0.833x – 2.667

or

y = 5/6x – 8/3

Example 2: For 1 point & slope

Determine the slope intercept form equation if the given slope of the line is 34, which passes through a point (2, 8)

Solution

Step 1:Take the given data of the line.

x₁ = 2, y₁= 8

m = 34

Step 2:Now put one point of the line and the slope of the line in the general expression of slope intercept form in order to find the y-intercept of the line.

y = mx + b

8 = 34 (2) + b

8 = 68 + b

8 – 68 = b

-60 = b

Step 3:Now place the slope of the line and y-intercept of the line to the general expression of slope intercept form to get the straight-line equation. 

y = mx + b

y = 34x + (-60)

y = 34x – 60

Or

y = 2(17x + 30)

The problems of finding straight-line equation using slope intercept form can also be solved by using a slope intercept calculator.

Solved through slope-intercept form calculator by AllMath

Example of Standard Form of a line

Determine the equation of the line by using the given coordinate points (2, -15) & (6, 5).

Solution

Step 1:Write the given points of the x-axis and y-axis.

x₁ = 2, x₂ = 6, y₁ = -15, y₂ = 5

Step 2:Now calculate the slope of the line in order to find the straight-line equation.

Slope of the line = m = [y₂ – y₁]/ [x₂ – x₁]

Slope = m = [5 – (-15)] / [6 – 2]

Slope = m = [5 + 15] / [4]

Slope = m = 20/ 4

Slope = m = 5

Step 3:Now write the general expression of point slope form and place the value of the slope of the line and a coordinate point on the line to get the straight-line equation.

y – y₁ = m (x – x_1­)

y – (5) = 5 * (x – 6)

y – 5 = 5 * x – 5 * 12

y – 5 = 5x – 60

y – 5x = -60 – 5

y – 5x = -65

Multiply the equation by -1 to make it in standard form

5x – y = 65

Conclusion

We’ve explored all the basics of straight-line equations in this article. The straight-line equation can easily be determined and represented in the form of point-slope form, slope intercept form, and standard form of a line.