2 3 Slope Intercept Form

Slope Intercept Form

The gradient intercept form of a straight line is one of the nigh common forms used to stand for the equation of a line. The slope intercept formula can be used to find the equation of a line when given the gradient of the straight line and the y-intercept( the y-coordinate of the point where the line intersects the y-centrality). Equation of line is the equation that is satisfied by each point that lies on that line. There are various methods to detect this equation of a straight line given as,

Slope-intercept form
Bespeak slope course
Ii-point course
Intercept class

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

1.	What is Slope Intercept Form of a Straight Line?
2.	Gradient Intercept Formula
three.	Derivation of Formula For Gradient Intercept Form
4.	Straight-Line Equation Using Slope Intercept Form
5.	Converting Standard Form to Gradient Intercept Course
half dozen.	FAQs on Slope Intercept Form

What is Slope Intercept Course of a Straight Line?

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

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

The decision of this equation is straightforward. To find the gradient intercept form of a directly line, nosotros would demand the slope, or the angle of inclination of this straight line from the ten-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 discover the equation of a line. For the gradient-intercept formula, we have to know the gradient of the line and the intercept cut by the line with the y-axis. Permit usa consider a straight line of gradient 'm' and y-intercept 'b'. The gradient intercept grade equation for a straight line with a slope, 'm', and 'b' every bit the y-intercept can be given every bit: y = mx + b.

Slope intercept form of a straight line

Gradient 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 = -ten + 4.
The equation of a line with slope (two) and passing through origin(y-intercept = 0) is given as: y = 2x.

Note: The slope of the line for which bending of inclination, θ is given can exist calculated as tan θ. Also, in the instance when we are given two points (x₁, y₁) and (10₂, y₂) lying on the straight line, the slope can be given as: (y_ii - y₁)/(10₂ - x₁). Let u.s.a. have a look at the slope-intercept formula and its derivation for a better understanding of the concept.

Gradient Intercept Formula

The slope-intercept formula is used to notice 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 detect the equation of a straight line. The slope-intercept formula is one of these formulas which is used when nosotros know the slope of the straight line, which is denoted past m, and the y-intercept of the direct line, which is denoted by b or (0, b). Let united states of america learn the gradient-intercept formula with a few solved examples. Hither is the slope-intercept formula.

Gradient Intercept Formula in Math

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

y = mx + b

where,

yard = the slope of the line
b = y-intercept of the line
(x, y) correspond every point on the line
ten and y have to be kept every bit the variables while applying the higher up formula.

Note: The slope-intercept formula cannot be applied to find the equation of a vertical line. Hither'due south an instance to understand the awarding of slope intercept formula.

Instance : The equation of a line is 3x + 4y + five = 0. Determine the slope and y-intercept of the line using the gradient intercept course.

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

4y = -3x - five
⇒ y = (-3/4)ten + (-5/iv)

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

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

Derivation of Formula For Slope Intercept Grade

Let usa consider a line whose slope is 'chiliad' that intersects the y-axis at (0, b), i.e., its y-intercept is b. Also, let the states consider an capricious point (x, y) on the line.

Slope-Intercept Formula Derivation

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

Using the slope formula, the gradient of a line joining ii points (x_one, y₁) and (x₂, y₂) is, m = (y₂ - y_ane)/(x₂ - ten_i)

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

Multiplying both sides by ten,
mx = y - b

Calculation '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 gradient intercept formula is derived.

Straight-Line Equation Using Slope Intercept Form

To find the equation of a line with an arbitrary inclination, nosotros would need 2 quantities: the inclination of the line (or its slope or the angle, θ, it makes with say, the 10-centrality) and the placement of the line (i.east. 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-centrality through which the line passes, or in other words, by specifying the y-intercept, b). Whatsoever line can exist determined uniquely using these two parameters.

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

Step one: Note down the y-intercept, 'b', and the slope of the line every bit 'yard'. Nosotros tin utilise the slope formula to discover the gradient of whatsoever direct line, in case it is not given directly and other relevant data is provided.
Step two: 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 signal (0, - 1). Find the equation of the line.

Solution: We accept, m = tan 60º = √3

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

Converting Standard Grade to Gradient Intercept Form

We tin can catechumen the equation of a line given in the standard form to gradient intercept form by rearranging and comparison. We know that the standard grade of the equation of a directly 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 gradient of the line and (-C/B) is the y-intercept.

Topics Related to Gradient Intercept Grade:

Euclidean Distance Formula
Geometry
ten and y axis

Important Notes on Gradient Intercept Form:

A line may have a negative slope in example the angle it makes with the positive x-direction is an birdbrained angle. The value of tan θ, in this case, will be negative, so m will be negative.
For whatever 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

Example 1: Using the slope intercept form, discover the equation of a straight line with slope 1/iii and whose y-intercept is (0, -5).

Solution:

To find the equation of the given line:

Given: the slope of the line is g = one/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 = (ane/3) ten - 5

Answer: The equation of the given line is, y = (1/three) ten - 5.
Case 2: Notice the equation of the horizontal line that intersects the y-axis at (0, three). Solve information technology using the gradient-intercept formula.

Solution:

To find the equation of the given line:

It is given that the y-intercept of the line is (0, b) = (0, 3) ⇒ b = iii.

Since the line is horizontal, its slope is m = 0.

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

y = mx + b

y = 0x + three

y = 3

Answer: The equation of the given line is, y = 3.
Instance three: Discover the equation of a line that is parallel to the line y = 3x - v and whose y-intercept is (-i/five).

Solution:

To notice: The equation of the line parallel to the given line.

Information technology is given that the y-intercept of the line is B = -1/v.

The equation of the given line is,

y = 3x - 5

Comparison this with y = mx + b, we get its gradient to be thousand = 3.

Since the given line is parallel to the required line, their slopes are equal.

Then the slope of the required line is, M = 3 as well.

Thus the equation of the required line using the gradient-intercept formula is,

y = Mx + B

y = 3x - 1/5

Answer: The equation of the required line is, y = 3x - 1/v.

go to slidego to slidego to slide

Breakdown tough concepts through simple visuals.

Math will no longer be a tough subject, particularly when you empathize the concepts through visualizations.

Book a Complimentary Trial Class

Practise Questions on Slope Intercept Grade

go to slidego to slide

FAQs on Slope Intercept Grade

What is Slope Intercept Form in Math?

The slope intercept form in math is one of the forms used to calculate the equation of a direct line, given the slope of the line and intercept it forms with the y-centrality. The slope intercept form is given as, y = mx + b, where 'k' is the slope of the direct line and 'b' is the y-intercept.

What is the Slope Intercept Form Equation?

The slope intercept equation is used to detect the full general equation of a straight line using its slope and the betoken where it intersects the y-axis. Slope intercept form equation is given as, y = mx + b.

How do y'all Find Slope-Intercept Form?

The slope intercept form of any line can be calculated merely using the slope and y-intercept. The gradient intercept form of a direct line is given equally,
y = mx + b
where,

(x, y) is an capricious point on the line
m is the slope of the line
b is the y-intercept

How to Detect the Equation of a Directly Line Using Slope Intercept Grade?

We demand the slope of the straight line and its point of intersection with the y-axis to find the direct-line equation using the slope intercept form. The slope of a line can be calculated using the gradient formula. Using the slope intercept form, equation of directly line tin can be calculated as, y = mx + b, where 'm' is the slope of the directly line and 'b' is the y-intercept.

What is Gradient-Intercept Formula?

The slope-intercept formula is one of the formulas used to discover the equation of a line. The gradient-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (10, y) is any bespeak on the line.

How To Derive the Slope-Intercept Formula?

Let us consider a line whose slope is m and whose y-intercept is (0, b). To find the equation of the line, consider a random point (ten, y) on information technology. Then using the slope formula, (y - b) / (10 - 0) = m. Solving information technology for y, we get y = mx + b.

What are the Applications of the Slope-Intercept Formula?

The slope-intercept formula is used to

find the equation of a line.
graph a line using the y-intercept and slope.
notice the gradient of a line easily.
find the intercepts of a line hands.

How to Find the Slope of a Line Using the Slope-Intercept Form?

Nosotros tin can find the gradient of a line using the slope-intercept form given equally, y = mx + b, where 'chiliad' is the slope of the line and 'b' is the y-intercept. Here is an example. Let us discover the slope of the line 6x - 3y = 5. Allow us solve this for 'y' to get into the slope-intercept form. Then we become y = 2x - (5/iii). Comparing this with the slope-intercept formula, y = mx + b, nosotros get its slope to be m = 2.

How to Catechumen Standard Form of Direct Line Equation to Slope Intercept Form?

The standard class of equation of a straight line is given as, Ax + Past + C = 0. Rearranging this standard course, we can observe the slope intercept of any straight line given in this form as, y = (-A/B)x + (-C/B), where (-A/B) makes the slope of the line and (-C/B) is the y-intercept.