A line passes through (2, −1) and (4, 5). Which answer is the equation of the line?

Straight lines can be represented in the form of linear equations on the cartesian plane. Let's solve a problem related to the concept of two-point form in straight lines.

Answer: The Equation of the Line that Passes through the Points (2, −1) and (4, 5) is 3x - y - 7 = 0.

Let's solve this step by step.

Explanation:

We are given: (x₁, y₁) = (2, −1) and (x₂, y₂) = (4, 5)

The two-point form of a line passing through these two points (x₁, y₁) and (x₂, y₂) is:

⇒ (y − y₁) = [(y₂ − y₁) (x − x₁)] / (x₂ − x₁)

Here, (y₂ − y₁) / (x₂ − x₁) is the slope of the line.

⇒ (y − y₁) (x₂ − x₁) = (y₂ − y₁) (x − x₁)

Substituting the values of points (x₁, y₁) and (x₂, y₂):

⇒ (y - (-1)) (4 - 2) = (5 - (-1)) (x − 2)

⇒ (y + 1) (2) = (6) (x - 2)

⇒ 2y + 2 = 6x - 12

⇒ 6x - 2y - 14 = 0

⇒ 3x - y - 7 = 0

Hence, the Equation of the Line that Passes through the Points (2, −1) and (4, 5) is 3x - y - 7 = 0.