In algebra, the term slope intercept form is a fundamental concept that is essential for understanding the equation of the line in a more precise way. The interpretation of the slope intercept form is a gateway to interpret and manipulate the linear equations.

The slope and the intercept of the line plays a crucial rule in the equation of slope intercept form to denote linear equation of the line. In this post, we’ll learn the basics and various aspects of the slope intercept form.

## What is Slope Intercept Form?

The slope intercept form is a fundamental concept in algebra that denotes the line equation in a unique manner. The form of this line equation would be represented as y = mx + b, in which x and y are the variables of linear equation of the line.

The term “m” represents the slope of the line, which is evaluated, with the help of run over rise of the coordinate points of the line. In addition, the term “b” represents the y intercept of the line which is the point where the line intersects the y-axis.

The term slope and y-intercept of the line are the essential components of the expression of the slope intercept form. You must have a sound knowledge about the slope and y-intercept of the line.

### What is slope?

The term slope is the inclination of the line which is also the steepness of the line. It is represented by the symbol “m”. The change in the variable of y over change in the variable of x will helps to find the slope of a line.

Such as change in the x variables is (x_{2} – x_{1}) and the change in the y variables (y_{2} – y_{1}) would be written as:

**Slope = m = (y _{2} – y_{1}) / (x_{2} – x_{1})**

The magnitude of the slope will let you know how steep the line is. The slope could be positive or negative.

**Positive slope**: It indicates an upward slant**Negative Slope**: It indicates a downward slant

### What is the y-intercept?

The term y intercept of the line is a point of the line which intersects y axis at any point. It is denoted by a symbol “b”. It indicates the value of y when the term x is equal to zero. It is very essential as it provides the starting point for plotting the line on graph.

Such as:

y = mx + b

set x = 0

y = m(o) + b

after simplification

**y = b**

By setting the equation at x = 0, it gives y = b as a reduce form which indicates that the value of y-intercept is y at x is zero.

## How to find the slope intercept form?

A y=mx+b calculator is a best way to find the slope intercept form with different techniques in seconds. Here are going to discuss some examples to learn how to find it manually.

### For Two Coordinate Points of Line

**Example 1:**

Determine the linear equation of the straight line through slope intercept form with the help of the given coordinate points of the line.

(x_{1}, y_{1}) = (-2, -6) & (x_{2}, y_{2}) = (8, 4)

**Solution**

**Step 1:** Take the x-axis and y-axis points of the line.

x_{1} = -2, x_{2} = 8, y_{1} = -6, y_{2} = 4

**Step 2:** Now evaluate the steepness of the line with the help of the formula of change in the variables of x and y axis through given points.

**General expression of the slope of the line **

Slope of the line = m = [y_{2} – y_{1}] / [x_{2} – x_{1}]

**Put the given values**

Slope = m = [4 – (-6)] / [8 – (-2)]

Slope = m = [4 + 6] / [8 + 2]

Slope = m = [10] / [10]

Slope = m = 5/5

Slope = m = 1

Hence, the slope of the line is positive which is an upward slant.

**Step 3:** Now put the above calculated slope of the line and the x and y axis point of the line to the expression of the slope intercept form to determine the y intercept of the line.

**y = mx + b**

-6 = 1(-2) + b

-6 = -2 + b

– 6 + 2 = b

-4 = b

**Step 4:** Now substitute the calculated slope of the line “m = 1” and the y intercept of the line “b = -4” to the expression of the slope intercept form to find the line equation.

**y = mx + b**

y = 1x + (-4)

y = x – 4

**Example 2:**

Determine the linear equation of the straight line through slope intercept form with the help of the given coordinate points of the line.

(x_{1}, y_{1}) = (8, 12) & (x_{2}, y_{2}) = (14, 10)

**Solution**

**Step 1:** Take the x-axis and y-axis points of the line.

x_{1} = 8, x_{2} = 14, y_{1} = 12, y_{2} = 10

**Step 2:** Now evaluate the steepness of the line with the help of the formula of change in the variables of x and y axis through given points.

**General expression of the slope of the line **

Slope of the line = m = [y_{2} – y_{1}] / [x_{2} – x_{1}]

**Put the given values**

Slope = m = [10 – 12] / [14 – 8]

Slope = m = [-2] / [6]

Slope = m = -1/3

Slope = m = -0.333

Hence, the slope of the line is negative which is a downward slant.

**Step 3:** Now put the above calculated slope of the line and the x and y axis point of the line to the expression of the slope intercept form to determine the y intercept of the line.

**y = mx + b**

12 = -1/3(8) + b

12 = -8/3 + b

12 + 8/3 = b

(36 + 8)/ 3 = b

44 / 3 = b

14.67 = b

**Step 4:** Now substitute the calculated slope of the line “m = -1/3” and the y intercept of the line “b = 44/3” to the expression of the slope intercept form to find the line equation.

**y = mx + b**

y = 1/3x + 44/3

y = (x + 44)/3

### For 1 point & slope

**Example 1:**

Find the linear equation of the line through slope intercept form with the help of the given coordinate point and slope of the line.

(x_{1}, y_{1}) = (12, 18) & slope = 10

**Solution**

**Step 1:** Take the given data of the line.

x_{1} = 12, y_{1} = 18

m = 10

**Step 2:** Now put the above slope of the line and the x and y axis point of the line to the expression of the slope intercept form to determine the y intercept of the line.

**y = mx + b**

18 = 10(12) + b

18 = 120 + b

18 – 120 = b

-112 = b

**Step 3:** Now substitute the slope of the line “m = 10” and the y intercept of the line “b = -112” to the expression of the slope intercept form to find the line equation.

**y = mx + b**

y = 10x + (-112)

y = 10x – 112

### For Slope and Y-intercept

**Example 1:**

Find the linear equation of the line through slope intercept form with the help of the given y intercept of the line and slope of the line.

b = -12 & slope = 15

**Solution**

**Step 1:** Take the given data of the line.

b = -12

m = 15

**Step 2:** Now substitute the slope of the line “m = 15” and the y intercept of the line “b = -12” to the expression of the slope intercept form to find the line equation.

**y = mx + b**

y = 15x + (-12)

y = 15x – 12

## Conclusion

The slope intercept form is very essential technique for writing the equation of the line in a well-known order as y = mx + b. There are various was to determine the slope intercept form equation such as two-point method, slope and point method, and slope and y intercept method.