Slope Intercept Form Formula

Slope Intercept Form

The slope intercept form of a straight line is one of the most common forms used to represent the equation of a line. The slope intercept formula can be used to find the equation of a line when given the slope of the straight line and the y-intercept( the y-coordinate of the point where the line intersects the y-axis). Equation of 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

Let us understand slope intercept formula, its derivation using solved examples.

What is Slope Intercept Form of a Straight Line?

The slope intercept form is a method used to determine the equation of a straight line in the coordinate plane. The equation of a straight line will be that relation which:

• the coordinates of any point on the line must satisfy.
• the coordinates of any point not on the line will not satisfy.

The determination of this equation is straightforward. To find the slope intercept form of a straight line, we would need the slope, or the angle of inclination of this straight line from the x-axis and the intercept that it makes with the y-axis.

Slope Intercept Form Definition

The slope-intercept form of a straight line is used to find the equation of a line. For the slope-intercept formula, we have to know the slope of the line and the intercept cut by the line with the y-axis. Let us consider a straight line of slope ‘m’ and y-intercept ‘b’. The slope intercept form equation for a straight line with a slope, ‘m’, and ‘b’ as the y-intercept can be given as: y = mx + b.

Slope Intercept Form Examples

Some examples of the slope intercept form are shown here.

• The equation of a line with slope (-1) and y-intercept (4) is found using: y = -x + 4.
• The equation of a line with slope (2) and passing through origin(y-intercept = 0) is given as: y = 2x.

Note: The slope of the line for which angle of inclination, θ is given can be calculated as tan θ. Also, in the case when we are given two points (x₁, y₁) and (x₂, y₂) lying on the straight line, the slope can be given as: (y₂ – y₁)/(x₂ – x₁). Let us have a look at the slope-intercept formula and its derivation for a better understanding of the concept.

Slope Intercept Formula

The slope-intercept formula is used to find the slope, the y-intercept, the x-intercept, or the equation of a straight line given the requisite parameters. There are different formulas available to find the equation of a straight line. The slope-intercept formula is one of these formulas which is used when we know the slope of the straight line, which is denoted by m, and the y-intercept of the straight line, which is denoted by b or (0, b). Let us learn the slope-intercept formula with a few solved examples. Here is the slope-intercept formula.

Slope Intercept Formula in Math

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

y = mx + b

where,

• m = the slope of the line
• b = y-intercept of the line
• (x, y) represent every point on the line
  x and y have to be kept as the variables while applying the above formula.

Note: The slope-intercept formula cannot be applied to find the equation of a vertical line. Here’s an example to understand the application of slope intercept formula.

Example : The equation of a line is 3x + 4y + 5 = 0. Determine the slope and y-intercept of the line using the slope intercept form.

Solution: We re-arrange the equation of the line to write it in the standard form y = mx + b.
We have:

4y = -3x – 5
⇒ y = (-3/4)x + (-5/4)

Thus, m = -3/4 , b = -5/4

Answer: The slope of the given straight line, m = -3/4 and the y-intercept, b = -5/4.

Derivation of Formula For Slope Intercept Form

Let us consider a line whose slope is ‘m’ that intersects the y-axis at (0, b), i.e., its y-intercept is b. Also, let us consider an arbitrary point (x, y) on the line.

Let us assume that (x₁, y₁) = (0, b) and (x₂, y₂) = (x, y).

Using the slope formula, the slope of a line joining two points (x₁, y₁) and (x₂, y₂) is, m = (y₂ – y₁)/(x₂ – x₁)

Using this formula, the slope of the above line is,
m = (y – b) / (x – 0)
⇒ m = (y – b) / (x)

Multiplying both sides by x,
mx = y – b

Adding ‘b’ on both sides,
y = mx + b

This is the general equation of a straight line involving its slope and its y-intercept. This form of the equation of the line is therefore termed the slope-intercept form. Hence, the slope intercept formula is derived.

Straight-Line Equation Using Slope Intercept Form

To find the equation of a line with an arbitrary inclination, we would need two quantities: the inclination of the line (or its slope or the angle, θ, it makes with say, the x-axis) and the placement of the line (i.e. where the line passes through with reference to the axes; we can specify the placement of the line by specifying the point on the y-axis through which the line passes, or in other words, by specifying the y-intercept, b). Any line can be determined uniquely using these two parameters.

The steps to find the equation of a line using the slope-intercept form are given below,

Step 1: Note down the y-intercept, ‘b’, and the slope of the line as ‘m’. We can apply the slope formula to find the slope of any straight line, in case it is not given directly and other relevant data is provided.
Step 2: Apply the slope intercept formula: y = mx + b.

Example: A line is inclined at an angle of 60° to the horizontal, and passes through the point (0, – 1). Find the equation of the line.

Solution: We have, m = tan 60º = √3

Thus, the equation of the line is, y = mx + c
⇒y = (√3)x + (−1)
⇒y = √3x − 1

Converting Standard Form to Slope Intercept Form

We can convert the equation of a line given in the standard form to slope intercept form by rearranging and comparing. We know that the standard form of the equation of a straight line can be given as, Ax + By + C = 0. Rearranging the terms to find the value of ‘y’, we get,
B × y = -Ax – C
⇒y = (-A/B)x + (-C/B),
where, (-A/B) makes the slope of the line and (-C/B) is the y-intercept.

Important Notes on Slope Intercept Form:

• A line may have a negative slope in case the angle it makes with the positive x-direction is an obtuse angle. The value of tan θ, in this case, will be negative, so m will be negative.
• For any line passing through the origin, the y-intercept will be (b = 0), so its equation will be of the form: y = mx.

Examples on Slope Intercept Form

What is the Slope Intercept Form of a Line?

The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers.

The slope, m, represents the steepness of a line. The slope of the line is also termed as gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.

Slope Intercept Form Equation

In this section, you will learn the derivation of the equation of a line in the slope-intercept form.

Consider a line L with slope m cuts the y-axis at a distance of c units from the origin.

Here, the distance c is called the y-intercept of the given line L.

So, the coordinate of a point where the line L meets the y-axis will be (0, c).

That means, line L passes through a fixed point (0, c) with slope m.

We know that, the equation of a line in point slope form, where (x₁, y₁) is the point and slope m is:

(y – y₁) = m(x – x₁)

Here, (x₁, y₁) = (0, c)

Substituting these values, we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c

Note: The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively.

Slope Intercept Form Formula

As derived above, the equation of the line in slope-intercept form is given by:

y = mx + c

Here,

(x, y) = Every point on the line

m = Slope of the line

c = y-intercept of the line

Usually, x and y have to be kept as the variables while using the above formula.

Slope Intercept Form x Intercept

We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:

y = m(x – d)

Here, 

m = Slope of the line

d = x-intercept of the line

Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

m = tan θ

Derivation of Slope-Intercept Form from Standard Form Equation

We can derive 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 of a straight line is:

Ax + By + C = 0

Rearranging the terms as:

By = -Ax – C

⇒y = (-A/B)x + (-C/B)

This is of the form y = mx + c

Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.

Slope Intercept Form Graph

When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form. Since it is in the form “y=”, hence it is easy to graph it or solve word problems based on it. We just have to put the x-values and the equation is solved for y.

The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.

Solved Examples

Example 1:

Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5).

Solution:

By the slope-intercept form we know;

y = mx+c

Given,

m = 3

As per the given point, we have;

y = -5 and x = -2

Hence, putting the values in the above equation, we get;

-5 = 3(-2) + c

-5 = -6+c

c = -5 + 6 = 1

Hence, the required equation will be;

y = 3x+1

Example 2:

Find the equation of the straight line that has slope m = -1 and passes through the point (2, -3).

Solution:

By the slope-intercept form we know;

y = mx+c

Given,

m = -1

As per the given point, we have;

y = -3 and x = 2

Hence, putting the values in the above equation, we get;

-3 = -1(2) + c

-3 = -2 + c

c = -3+2 = -1

Hence, the required equation will be;

y = -x-1

Example 3:

Find 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 -5 

(ii) x-intercept is 7/3

Solution:

Given, tan θ = 1/2

So, slope = m = tan θ = 1/2

(i) y-intercept = c = -5

Equation of the line using slope intercept form is:

y = mx + c

y = (1/2)x + (-5)

Or

2y = x – 10

x – 2y – 10 = 0

(ii) x-intercept = d = 7/3

Equation of slope intercept form with x-intercept is:

y = m(x – d)

y = (1/2)[x – (7/3)]

Or

2y = (3x – 7)/3

6y = 3x – 7

3x – 6y – 7 = 0

Practice Problems

1. Find the slope of the line y = 5x + 2.
2. Find the slope of the line which crosses the line at point (-2,6) and have an intercept of 3.
3. What is the equation of the line whose angle of inclination is 45 degrees and x-intercept is -⅗?
4. Write the equation of the line passing through the point (0, 0) with slope -4.

Slope-Intercept Form: Definition & Examples

The slope-Intercept form is among the four other methods to determine the equation of any straight line. Straight lines are generally represented as y = mx+c. This standard equation of a straight line defines that the x and y variables have a maximum power of unity. This means that x = x1 and y = y1.

Also, this confirms that the slope-intercept form can be applied only to linear equations. You should be familiar with two-variable linear equations. You should be aware that the graph of such equations is a straight line. You should also be confident about what are the y-intercept, x-intercept, and slope characteristics of linear equations.

If you try to solve an equation like x = 2a³ + y, it might turn a bit tricky, and there’s a probability of getting errors by slope-intercept form.    

The other four forms or methods of solving a linear equation are listed below:

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

 In this article, you will learn the slope-intercept form, what is slope-intercept form, slope-intercept form equation, how to find slope-intercept form, how to write an equation in slope-intercept form, and the slope-intercept form of a linear equation.

What is Slope-Intercept Form?

A straight line’s slope-intercept form is amongst the most frequent ways to describe its equation. Consider you are given the slope of a line, and you know that line crosses the y-axis at some point in the cartesian plane. In such cases, it is wise to use the slope-intercept form.

You can divide the slope-intercept form into two concepts:

• The slope of a line: The ratio of the difference between the coordinates of the y-axis to the difference between the coordinates of the x-axis.
• ‘y’ Intercept: The point on the y-axis where the line with a certain slope crosses or intersects is the y-intercept.  

Note: The coordinates of the y-intercept are always (0, y). This is because the line whose equation has to be determined always cuts the y-axis at x = 0.

Slope Intercept Formula 

The slope-intercept equation is written as: 

Y = mx + b 

Where  

x, y are the x and y coordinates, 

m is the slope of the line, and  

b is the y-intercept. 

What is the Slope-Intercept Form of a Straight Line?

Have a careful look at the figure mentioned below. You will see a straight line ‘AB’ which passes through the 1st quadrant of the coordinate system and cuts the y-axis at point C. The coordinates of this point C are, let’s say, C (0, y). Also, if we see the line ‘ABC’ it inclines some degrees from the x-axis. This is the slope of the given straight line. These are the only things we require to find the equation of a straight line using the slope-intercept form. 

Note: If the coordinates of the lines satisfy or agree with the equation, then the coordinates are correct. If the coordinates don’t satisfy the equation, then they are not the coordinates of that line. 

Slope Intercept Form Equation 

Now, we are confirmed that the slope-intercept form of a straight line is a neat and simplified way to find the equation of a line. In mathematics, the slope-intercept formula is given as :

y = mx + k, or you can use any variable in place of these terms, but remember that:

1. ‘y’ and ‘x’ always remain unchanged. They are the reason why this term is an equation.
2. ‘m’ is defined as the slope of the line, and 
3. ‘k’ is the ‘y’ intercept which has been talked about earlier. 

Some Examples of Slope-Intercept Form

To ease all the theoretical knowledge mentioned above, let us look at some of the examples of slope-intercept form and learn how to write an equation in slope-intercept form.

Example 1: The slope of a line XY is (-1), and the y-intercept is (10). What is the equation of the line?

Answer: y = (-1) x + 10.

Example 2: The slope of line XY is (7), and the line passes from the origin. What is the equation of the line?

Answer: y = (7) x + 0 => y = 7x {Because when a line passes from the origin the x and y-intercept are (0,0)}

Example: 

If the y-intercept is 8 and the slope is 3, find the straight-line equation. 

Solution: 

Step 1: Determine the values. 

b = 8 

m = 3 

Step 2: Construct the slope form equation and enter the values. 

y = mx + b 

y = 3x + 8 

Example: 

Using the slope-intercept form, determine the equation of a straight line with a slope of 1/7 and a y-intercept of (0, -9). 

Solution: 

To find the equation of a given line, perform the following steps: 

Given: the line’s slope is m = 1/7. 

The line’s y-intercept is (0, b) = (0, -9) b = -9. 

The equation of the given line is calculated using the slope-intercept formula. 

y = mx + b 

y = (1/7) x – 9 

Answer: The given line’s equation is y = (1/7) x – 9. 

How to find the Slope of a line?

You might find this one a tricky concept, but once you understand it, you will excel in finding the slopes of straight lines.

Method 1: If you know at what angle the line is inclined from the x-axis, you can find the slope by using simple trigonometry. Let us say that our line is inclined at an angle of ‘a’ from the x-axis then,

Slope ‘m’ = tan (a) 

Method 2: If the coordinates of two points on the same line are given, we can easily find the slope by finding the ratio of the difference between the coordinates of the y-axis to the difference between the coordinates of the x-axis. For example: Let us say points (x, y) and (x1, y1) are in the same line, then 

Slope ‘m’ = (y – y1) / (x – x1)

For a deeper grasp of the notion, see the slope-intercept formula and its derivation below.

Derivation of Formula For Slope-Intercept Form

Let us assume a random arbitrary point on line: A (x, y). Also, let us consider that the slope of the line is ‘m’, and this line intersects the vertical axis (y-axis) at k such that point C is (0, k). See the diagram for better clarification.

Utilizing the formula for finding the slope when two points are given we have:

m = (y – y1) / (x – x1)

m = (y – k) / (x – 0)

m (x – 0) = y – k

mx = y – k

y = mx + k

This is a generic equation for a straight line that includes the slope and y-intercept. The slope-intercept form is the name given to this type of line equation. As a result, the slope-intercept formula is created. Now you know how to write slope-intercept form.

It is important to note that the slope-intercept formula cannot be used to get the equation of a vertical line. This is because a vertical line has no y-intercept.

Straight-Line Equation Using Slope Intercept Form

The two necessary parameters, that is, the slope ‘m’ of a line, and the y-intercept ‘k’, are utilized to establish the uniqueness of any line.

The procedures for determining a line’s equation using the slope-intercept form are outlined below.

Step 1: Write down the y-intercept, ‘k,’ and the line’s slope, ‘m.’ If the slope of a straight line is not supplied explicitly and other necessary data is available, we may use the slope formula to find it. (The slope formulas are mentioned above in the article) 

Step 2: Use the slope-intercept formula to solve the problem: y = MX + k.

 Example: A line is inclined at an angle of 45° to the x-axis, and passes through the point (0, 10). Find the equation of the line.

Answer: We have, m = tan 45º = 1

Thus, the equation of this line is, y = mx + k

y = (1) x + (10)

y = x − 10

Isn’t the slope-intercept form of a linear equation quite simple for you?

Now that we have covered all the theoretical aspects of this topic, let us look at some decent examples to ace the concepts of this article.

Examples of Slope-Intercept Form

Example 1: Given the slope of a line is 1/2, and it crosses the y-axis at (0, -3). Find the equation of the line.

Solution: We are given slope ‘m’ = 1/2,

Coordinates of a point on the y-axis where the line crosses = (0, -3) 

Therefore k = -3

Equation of the line => y = mx + k,

y =(1/2) x + (-3)

y = x/2 – 3

Answer: The equation of the line is y = x/2 – 3.

Example 2: Find the equation using the horizontal line slope intercept formula that intersects the y-axis at (0, 8). Solve it.

Solution: We are given that the line is horizontal, which means the angle of inclination ‘a’ is 0.

Thus tan (0) = 0

Coordinates of a point on the y-axis where the line crosses = (0, 8) 

Therefore k = 8

Equation of the line => y = mx + k,

y = 0 x + (8)

y = 8

Answer: The equation of the line is y = 8. That sums up the article about slope-intercept form, how to find slope-intercept form, how to write an equation in slope-intercept form, and the slope-intercept form of a linear equation. If you still have doubts and think you have missed some concepts, then you can surf back to this article and find the answers to your queries in section-wise elaborated paragraphs. Moreover, do not forget to learn new concepts and practice them regularly.

Frequently Asked Questions

1. What is Slope Intercept Form in Math?

Ans. In math, slope intercept form is a way of writing an equation for a line in the form y = mx + b. The m represents the slope of the line and the b represents the y-intercept. The slope intercept form is used when you want to find either a point on a line or solve for y if you know x.

2. What is the Slope Intercept Form Equation?

Ans. The slope intercept form equation is y = mx + b. This equation is used to find the slope and y-intercept of a line. The slope represents how fast the line changes from point to point, while b represents the starting point of the line (i.e., where it intersects with the y-axis).

3. How do you Find Slope-Intercept Form?

Ans. To find the Slope Intercept form, you need to know what the slope of a line is. The slope of a line can be calculated by: slope = (y2 – y1)/(x2 – x1)

4. What is the Slope-Intercept Formula?

Ans. The slope intercept formula is a simple equation that can be used to find the slope and y-intercept of a line.

The slope intercept formula is:

y = mx + b

5. How To Derive the Slope-Intercept Formula?

Ans. To derive the Slope Intercept Formula, we must first recall the definition of a line. A line is a set of points in two-dimensional space that satisfy an equation of the form:

y = mx + b

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