# Finding The Equation Of The Line By Using The Slope-Intercept Form And Point-Slope Form

In mathematics and physics, equations of the line are used widely. The equation of the line is essential in different branches of science and everyday problems. It is that equation in which the degree of each variable is one. This article will discuss the Slope-Intercept And Point-Slope of Linear Equations.

The equation of the line can be calculated by using different methods. In this post, we will discuss two helpful methods for calculating the equation of the line. These methods are a slope-intercept and point-slope form of Linear Equations.

## What are slope-intercept form and point-slope form?

The slope-intercept form and the point-slope forms of linear equations are the methods or techniques used to determine the equation of the line. Both methods follow different ways to calculate the line equation. Both methods have one thing in common. They used the slope of the line to evaluate the equation of the line.

The first method to find the equation of the line is a slope-intercept form. It follows an equation such as **y = mx + b; in** this equation,** x** and **y** are variables, m is the slope of the line, and **b** is the y-intercept form. If you want to use this method, first of all, you have to calculate the slope of the line by using the given points of the line.

The point-slope form is the other method to calculate the line equation. This method follows an equation like **y – y _{1} = m (x – x_{1}),** in which

**x**and

**y**are variables, m is the slope of the line, and

**x**and

_{1}**y**are the coordinate points of the line.

_{1}Like the above method, you have to calculate the slope of the line using a point-slope form equation to evaluate the equation of the line. Use a point-slope form calculator to get the result of your question according to the above equation.

The equations of both methods are used to find the exact line equation.

### How to calculate the equation of the line by using the above methods?

For the calculation of the equation by using the above methods, let us take some examples using the equations of these methods.

#### Example 1: By Slope-intercept form

Use (13, -12) and (19, 9) points to calculate the equation of the line by using the slope-intercept form.

**Solution using the slope-intercept form:**

**Step 1:** First of all, give names to the given points.

X_{1} = 13, x_{2} = 19, y_{1} = -12,
y_{2} = 9

**Step 2:** Now, use the above points to calculate the slope of the line.

Slope = m = y_{2} – y_{1} / x_{2} –
x_{1}

Slope = m = 9 – (-12) / 19 – 13

Slope = m = 9 + 12 / 6

Slope = m = 21/6

Slope = m = 3.5

**Step 3:** Now, take the general equation of the slope-intercept form.

y = mx + b

**Step 4:** Use any pair of points to get the y-intercept of the line.

y = mx + b

19 = 3.5(9) + b

19 = 31.5 + b

19 – 31.5 = b

-12.5 = b

b = -12.5

**Step 5:** Now put the calculated y-intercept and the slope
in the slope-intercept form equation.

y = mx + b

y = 3.5x – 12.5

This is the required equation of the line.

#### Example 2: By Slope-intercept form

Use (-3, 2) and (-9, 33) points to calculate the equation of the line by using a slope-intercept form.

**Solution using the slope-intercept form:**

**Step 1:** First of all, give names to the given points.

X_{1} = -3, x_{2} = -9, y_{1} = 2, y_{2}
= 33

**Step 2:** Now, use the above points to calculate the slope of the line.

Slope = m = y_{2} – y_{1} / x_{2} –
x_{1}

Slope = m = 33 – 2 / -9 – (-3)

Slope = m = 31 / -9 + 3

Slope = m = 31/-6

Slope = m = -5.17

**Step 3:** Now, take the general equation of the slope-intercept form.

y = mx + b

**Step 4:** Use any pair of points to get the y-intercept of the line.

y = mx + b

2 = -5.17(-3) + b

2 = 15.51 + b

2 – 15.51 = b

-13.51 = b

b = -13.51

**Step 5:** Now put the calculated y-intercept and the slope
in the slope-intercept form equation.

y = mx + b

y = -5.17x – 13.51

This is the required equation of the line.

#### Example 3: By point-slope form

Use (23, -2) and (29, 13) points to calculate the equation of the line by using a point-slope form.

**Solution using the point-slope form:**

**Step 1:** First of all, give names to the given points.

X_{1} = 23, x_{2} = 29, y_{1} = -2,
y_{2} = 13

**Step 2:** Now, use the above points to calculate the slope of the line.

Slope = m = y_{2} – y_{1} / x_{2} –
x_{1}

Slope = m = 13 – (-2) / 29 – 23

Slope = m = 13 + 2 / 6

Slope = m = 15/6

Slope = m = 5/2

Slope = m = 2.5

**Step 3:** Now, take the general equation of the point-slope form.

y – y_{1} = m (x – x_{1})

Step 4: Put the calculated slope and any pair of sets.

y – y_{1} = m (x – x_{1})

y – (-2) = 2.5(x – 23)

y + 2 = 2.5x – 57.5

y + 2 – 2.5x + 57.5 = 0

y – 2.5x + 59.5 = 0

2.5x – y – 59.5 = 0

#### Example 4: By point-slope form

Use (43, -21) and (129, 13) points to calculate the equation of the line by using a point-slope form.

**Solution using the point-slope form:**

**Step 1:** First of all, give names to the given points.

X_{1} = 43, x_{2} = 129, y_{1} = -21,
y_{2} = 13

**Step 2:** Now, use the above points to calculate the slope of the line.

Slope = m = y_{2} – y_{1} / x_{2} –
x_{1}

Slope = m = 13 – (-21) / 129 – 43

Slope = m = 13 + 21 / 86

Slope = m = 34/86

Slope = m = 17/43

Slope = m = 0.3953

**Step 3:** Now, take the general equation of the point-slope form.

y – y_{1} = m (x – x_{1})

Step 4: Put the calculated slope and any pair of sets.

y – y_{1} = m (x – x_{1})

y – (-21) = 0.3953(x – 43)

y + 21 = 0.3953x – 22.7298

y + 21 – 0.3953x + 22.7298 = 0

y – 0.3953x + 43.7298 = 0

0.3953x – y – 43.7298 = 0

##### Summary

These two methods are more frequently used to evaluate the equation of the line. From this post, you can grab the basic concepts of these methods and easily solve the problems related to this topic.