Equation of a Straight Line in One Point Form

The equation of a straight line can be expressed in various forms, each useful in different scenarios. One such form is the "point-slope" form, also known as the "one point form," which is particularly convenient when we know one point on the line and the slope of the line.

Point-Slope Form

The point-slope form of the equation of a straight line is given by:

[ y - y_1 = m(x - x_1) ]

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line.

Derivation

The point-slope form is derived from the definition of the slope of a line. The slope is the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. If we take $(x_1, y_1)$ as a known point and $(x, y)$ as any other point on the line, the slope $m$ can be expressed as:

[ m = \frac{y - y_1}{x - x_1} ]

Rearranging this equation gives us the point-slope form:

[ y - y_1 = m(x - x_1) ]

Important Points

Aspect	Description
Slope ($m$)	A measure of the steepness of the line.
Known Point ($(x_1, y_1)$)	A point through which the line passes.
Variables ($x$ and $y$)	Represent any point on the line.
Applicability	Useful when the slope and one point on the line are known.

Examples

Example 1: Finding the Equation of a Line

Suppose we have a line with a slope of $2$ that passes through the point $(3, -1)$. To write the equation of this line in point-slope form, we simply plug these values into the formula:

[ y - (-1) = 2(x - 3) ]

Simplifying, we get:

[ y + 1 = 2x - 6 ]

Or:

[ y = 2x - 7 ]

Example 2: Using the Equation to Find Another Point

Given the equation of a line in point-slope form:

[ y - 4 = -3(x - 2) ]

We can find another point on the line by choosing a value for $x$ and solving for $y$. Let's choose $x = 5$:

[ y - 4 = -3(5 - 2) ]

[ y - 4 = -3 \cdot 3 ]

[ y - 4 = -9 ]

[ y = -5 ]

So, the point $(5, -5)$ lies on the line.

Converting to Other Forms

The point-slope form can be easily converted to other standard forms of the equation of a line, such as the slope-intercept form ($y = mx + b$) or the standard form ($Ax + By = C$).

Conversion to Slope-Intercept Form

Starting with the point-slope form:

[ y - y_1 = m(x - x_1) ]

We can solve for $y$ to get the slope-intercept form:

[ y = mx - mx_1 + y_1 ]

Here, $b = -mx_1 + y_1$ is the y-intercept of the line.

Conversion to Standard Form

To convert to standard form, we rearrange the point-slope form to get:

[ y - mx = y_1 - mx_1 ]

And then write it as:

[ mx - y = mx_1 - y_1 ]

If necessary, we can multiply through by -1 to ensure that $A$ is positive:

[ -mx + y = -mx_1 + y_1 ]

The equation of a straight line in one point form is a powerful tool in coordinate geometry. It allows for quick and efficient calculations when dealing with straight lines, especially when the slope and a single point on the line are known. By understanding and applying this form, one can solve a variety of problems related to straight lines in mathematics.