Slope Intercept Form: Formula, Equation, Derivation & Graph

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Slope intercept form,In mathematics, is a way to express the equation of a line. It's also the most frequently used way to express the equation of a line. When a straight line intersects a segment on the axis, it is called an intercept.

To be able to use slope intercept form, you'll need ‘slope of the line’ and ‘y- intercept of the line’. The variable 'm' represents the slope of a straight line and is utilised in the equation

y = mx+b.

Table of Content

Slope intercept form formula
Slope Intercept Form of an Equation
Slope Intercept Form Graph
Slope Intercept Form x Intercept
Slope Intercept Form Derivation
Things to Remember
Sample Questions

Key Takeaways- Slope intercept form, straight line, x-intercept, y-intercept, slope-intercept, slope

Also read- Definite Integral Formula

Slope Intercept Form Formula

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The formula for slope-intercept form is y = mx + b, here-

m represents the slope of the line.
b represents the y-value of the line’s y-intercept.
x, y represent every point on the line.

The equation of a vertical line cannot be found through the application of the slope-intercept formula.

Slope intercept form formula

Also read- Planes

Slope Intercept Form of an Equation

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The formula for slope-intercept form is y = mx + b. Let us see how to derive the slope-intercept form equation of a straight line.

Step 1: Let L be a line with slope m and y-intercept b. Highlight the point that should be on the line. Justify your choice.

(b, 0) (0, b) (0, m) (m, 0)

The coordinate of x is 0 in the point that includes the y-intercept.

Step 2: Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b), and another point (x, y) on the line.

Slope m = change in y-values / change in x-values

Slope m = (y - b) / (x - 0)

Slope m = (y - b) / x

Step 3: Mostly y is written on one side of the equationIn an equation of a line. Solve the equation from Step 2 for y.

m = (y - b) / x

Multiply both sides by x

m.x = [(y - b) / x].x

mx = y - b

Add b to both sides of the equation.

mx + b = (y - b) + b

mx + b = y

Write the equation with y on the left side.

y = mx + b

Also read- Important formula

Slope Intercept Form Graph

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We obtain a straight line when we draw the graph for the slope-intercept form equation. The optimum form is slope-intercept. It is simple to graph or answer word problems based on it since it is in the form "y=". All we have to do now is plug in the x-values, and the equation for y is solved.

The best thing about the slope-intercept form is that we can get the slope and intercept values right out of the equation.

Also read- Algebra Formula

Slope Intercept Form x Intercept

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We can use the slope-intercept form (y = mx + b). here m and b are constants.

m = slope of the line

b = x- intercept of line

Setting y = 0 will give us the x-value at which the line crosses the x-axis, which is the x-intercept.

The slope of a line can also be expressed as tangent angle such as

m = tan θ

Slope Intercept Form Derivation

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The slope intercept form can be derived from the straight line equation in standard from as given below:

The straight line standard form of an equation can be written as:

Ax + By + C = 0

The equation can be rearranged as: By = -Ax - C

y = (-A/B)x + (-C/B)

This is of the form y = mx + c

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

Things to remember

Straight lines are produced by linear functions.
The slope-intercept formula is used when we know the slope of the straight line.
For a line passing through the origin, the y-intercept will be (b = 0), so the equation will be of the form: y = mx.
The slope-intercept formula cannot be applied to find the equation of a vertical line.

Also read: Isosceles Triangle Theorems

Sample Question

Ques- Using the slope intercept form, find the equation of a straight line with slope 1/3 and whose y-intercept is (0, -5). [2 marks]

Ans: To find the equation of the given line:

Given: the slope of the line is m = 1/3.

the y-intercept of the line is (0, b) = (0, -5) ⇒ b = -5.

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

y = mx + b

y = (1/3) x - 5

The equation of the given line is, y = (1/3) x - 5.

Ques- What is the y-intercept for the following equation:y + 84x = 157x + 250 [2 marks]

Ans: The y-intercept for an equation can be found through two ways. You could substitute 0 in for x.

This would give you:y + 84 ∗ 0= 157 ∗ 0 + 250.

you get: y = 250

However, this can also be done by finding the slope-intercept form of the line. You do this by solving for y. Indeed, this is very, very easy. Recall that the slope intercept form is:

Y = mx + b

This means that, as written, your equation obviously has b = 250.

Ques- Find the slope and y-intercept of the straight-line 4x - 7y + 1 = 0. [2 marks]

Ans: The equation of the given straight line is

4x - 7y + 1 = 0

⇒ 7y = 4x + 1

⇒ y = 4/7x + 1/7

Now, compare the above equation with the equation y = mx + b we get,

m = 4/7 and b =1/7.

Therefore, the slope of the given straight line is 4/7 and its y-intercept = 1/7 units.

Ques- With the slope 6 and y-intercept 4, find the equation of a line? [2 marks]

Ans: Here,

m = 6 and

b = 4

With the formula: y = mx + b

The equation of the line will be y = 6x + 4

Ques- Write the equation of the line in slope-intercept form where slope is -2 and passing through the point (0, 9). [2 marks]

Ans: Given,

Slope = m = -2

Point = (x, y) = (0, 9)

Equation of a line in slope-intercept form is:

y = mx + b….(i)

According to the given,

9 = (-2)(0) + b

b = 9

Substituting the value of m and b in equation (i),

y = -2x + 9

This is the required equation of a line in slope-intercept form.

Ques- Find the slope and y-intercept of the equation of line 2x – y + 5 = 0. [2 marks]

Ans: Given equation of a line:

2x – y + 5 = 0

Thus,

y = 2x + 5

This is of the form y = mx + b

Here, m = 2 and b = 5

Therefore, slope = 2 and y-intercept = 5

Ques- What is the y-intercept of the line with the following equation 2y− 4x = 10x − 20 [3 marks]

Ans:There are two ways that you can find the y-intercept for an equation. You could substitute 0 in for x. This would give you: 2y − 4∗ 0 = 10∗ 0 − 20

Simplifying, you get:

2y = -20

y = -10

Another way to do this is by finding the slope-intercept form of the line. You do this by solving for

2y = 14x − 20

Just divide everything by 2

y = 7x - 10

Remember that the slope-intercept form gives you the intercept as the final constant. Hence, it is -10

Ques- Find the equation of the line whose slope is 8 and the coordinates of the point are (3, 5). [2 marks]

Ans: Given, m = 8

(x1, y1) = (3, 5)

The formula for the equation of a line is,

y – y1 = m (x – x1)

y – 5 = 8 (x – 3)

y – 5 = 8x – 24

8x – y = 19

8x – y – 19 = 0

Ques- Form the equation of the straight line that runs parallel to another straight line of equation 2x + 3y + 11 = 0. The sum of the intercepts intercepted by the axis is 15. [3 marks]

Ans: If 2x + 3y + 11 is the equation of the line, then the equation parallel to this line will be 2x + 3y + c = 0

According to the question, a + b = 15 if the equation x/a + y/b =1 is considered, where a and b are intercepts cut on x as well as y axis.

Let’s rearrange 2x + 3y + c = 0 to bring it in x/a + y/b = 1 form.

2x + 3y = -c

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

Hence, a = -c/2 and b = -c/3

Therefore,

-c/2 + (-c/3) = 15

-3c/6 - 2c/6 = 15

-5c/6 = 15

c = -18

So, the equation of the line is: 2x + 3y - 18 = 0

Ques- What are the coordinates of the foot of the perpendicular to the point with coordinate (-1,3) to the line with equation 3x - 4y - 16 = 0. [3 marks]

Ans: Let us assume that (a,b) are the coordinates of the foot of the perpendicular from the point (-1,3) to the line with equation 3x - 4y - 16 = 0.

Slope of the line (m1) is (b-3) / (a+1)

Slope of the line (m2) is ¾

As these lines are perpendicular, m1.m2 = -1

{(b-3) / (a+1)} . (¾) = -1

=> 4a + 3b = 5 …(1)

(a,b) lies on 3x - 4y = 16

Therefore, 3a - 4b = 16 …(2)

a = 68/25

b = -49/25

Thus, the coordinates are {68/25 , 49/25}

Ques- The equation of a given line is 3x - 7y + 1. Calculate the slope. [2 marks]

Ans: According to formula, we know that,

y - y1 = m (x - x1)

Therefore, x1 = 3, and y1 = -7

Slope of the line = -(3/-7)

Hence, the slope of the given line is 3/7.

Mathematics Related Links:

Slope of line	X and y intercepts
Collinear points	Linear equation
Graph of a Linear Equation in Two Variables	Curve
Linear Equation in Two Variables	Pair of Linear Equations in Two Variables Formula
Substitution Method of Solving a Pair of Linear Equations	Equations Reducible To Linear Form

Slope Intercept Form: Formula, Equation, Derivation & Graph

Slope Intercept Form Formula

Slope Intercept Form of an Equation

Slope Intercept Form Graph

Slope Intercept Form x Intercept

Slope Intercept Form Derivation

Things to remember

Sample Question

CBSE CLASS XII Related Questions

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

6.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

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Slope Intercept Form: Formula, Equation, Derivation & Graph

Slope Intercept Form Formula

Slope Intercept Form of an Equation

Slope Intercept Form Graph

Slope Intercept Form x Intercept

Slope Intercept Form Derivation

Things to remember

Sample Question

CBSE CLASS XII Related Questions

1. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

3. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

6.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

2.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

6.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).