Understanding and Deriving the Point Slope Equation | A Guide to Linear Equations on a Cartesian Plane

Point Slope Equation

The point slope equation is a linear equation that represents a line on a Cartesian plane

The point slope equation is a linear equation that represents a line on a Cartesian plane. It is typically expressed in the form y – y₁ = m(x – x₁), where (x₁, y₁) represents a point on the line, and m represents the slope or gradient of the line.

Here’s how the point slope equation is derived:

1. Start with the slope-intercept form of a linear equation, which is y = mx + b. In this form, m represents the slope, and b represents the y-intercept.

2. Choose a point (x₁, y₁) on the line. Let’s say this point is (x₁, y₁).

3. Substitute the values of x₁ and y₁ into the slope-intercept form. We get y₁ = mx₁ + b.

4. Rearrange the equation to isolate y on the left side. This gives us y₁ – mx₁ = b.

5. Notice that y – y₁ represents the change in y-coordinate from the given point. Similarly, x – x₁ represents the change in the x-coordinate.

6. Replace y₁ with y, x₁ with x, and b with -m(x₁), and we obtain the point slope equation: y – y₁ = m(x – x₁).

The point slope equation is useful for a variety of applications, such as finding the equation of a line passing through a given point with a known slope or finding the equation of a line when multiple points are given. It provides a straightforward way to represent a line algebraically based on a point and its slope.

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