A straight line in a cartesian plane passes through an infinite number of points. Each of these points has its own x and y- coordinates. The points a line passes through are used to find its slope. Not only that but such points can also be used to write the equation of a line. One such method is discussed below.

Point Gradient formula

Out of a lot of methods to write/ find/ express the equation of a straight line in a cartesian form, the point-slope or point-gradient formula holds a very significant place in coordinate geometry. As its name suggests, this form of the equation consists of one point that the line passes through and its slope. 

Formula

The point-gradient formula is given as follows:

y – y₁ = m(x – x₁)

Where,

• x and y depict general point coordinates.
• x₁ and y₁ are the numerical coordinates of a point through which the line passes.
• m represents the slope of the given line.

Derivation of point-gradient formula

Slope of a line passing through two points (x, y) and (x₁, y₁) = m = 

Multiplying both sides by (x – x₁), we have:

⇒ y – y₁ = m(x – x₁)

Hence proved.

Sample Problems

Question 1: What is the equation of a line passing through (2, −4) and slope 5?

Solution:

The given point is (2, −4). Thus, x₁ = 2, y₁ = −4.

Also, m = slope = 5. 

We know the point slope equation of a line is given by y – y₁ = m(x – x₁).

Substituting the above values in the equation, we have:

y – (-4) = 5(x – 2)

⇒ y + 4 = 5x − 10

⇒ y = 5x − 10 − 4

⇒ y = 5x − 14

Question 2: What is the equation of a line passing through (5, 2) and slope 3/4?

Solution:

The given point is (5, 2). Thus, x₁ = 5, y₁ = 2.

Also, m = slope = 3/4.

We know the point slope equation of a line is given by y – y₁ = m(x – x₁).

Substituting the above values in the equation, we have:

y − (2) = 3/4(x − 5)

⇒ y − 2 = 3x/4 − 15/4

⇒ y = 3x/4 − 15/4 + 2

⇒ y = 3x/4 − 7/4

Question 3: What is the equation of a horizontal line passing through (3, 3)?

Solution:

The given point is (3, 3). Thus, x₁ = 3, y₁ = −3.

Since the slope of a horizontal line is zero, m = 0.

We know the point slope equation of a line is given by y – y₁ = m(x – x₁).

Substituting the above values in the equation, we have:

y − (3) = 0(x − 3)

⇒ y – 3 = 0

⇒ y = 3

Question 4: It is given that a line passes through the points (1, 1) and (-2, 4). Find its equation using the point-slope formula.

Solution:

We know the point slope equation of a line is given by y – y₁ = m(x – x₁).

In order to use the point- slope form, we need to calculate the slope of the line first.

Slope = m = 

⇒ m = -1

Substituting the above values in the equation, we have:

y – (1) = -1(x – 1)

⇒ y − 1 = -x + 1

⇒ y = -x + 2

Question 5: What is the equation of a line passing through (0, 3) and slope 8?

Solution:

The given point is (0, 3). Thus, x₁ = 0, y₁ = 3.

Also, m = slope = 8.

We know the point slope equation of a line is given by y – y₁ = m(x – x₁).

Substituting the above values in the equation, we have:

y − 3 = 8(x – 0)

⇒ y − 3 = 8x

⇒ y = 8x + 3