91Ó°ÊÓ

Understanding the equation of a line is crucial in geometry and algebra. It represents all the points that lie on a straight line in a 2-dimensional space. The most common forms of the equation of a line are the **point-slope form** and the **slope-intercept form**.
To write the equation of a line, two critical components are required: the **slope** and a **point** on the line. The slope tells us how steep the line is, while the point anchors the line in its position on the graph.
For example, if we want to find a line parallel to another line such as the one given by the equation \( y = -34x + 1 \), we start with knowing that parallel lines must have the same slope. Therefore, any line parallel to it must also have a slope of \(-34\).

The point-slope form is highly useful when you know the slope of a line and one point on it. It's represented by the formula:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) represents the slope, and \((x_1, y_1)\) are the coordinates of the given point. This formula is a convenient way to directly plug in known values and quickly find the line's equation.
For instance, given that the slope \(m = -34\) and the point \((x_1, y_1) = (4, 1/4)\), we substitute these into the point-slope form:

• \( y - 1/4 = -34(x - 4) \)

This step not only provides the line in a usable format but also connects the known values in a clear and structured way.

The slope-intercept form simplifies the equation to make it easier to understand the line's slope and y-intercept at glance. It is written as:

• \( y = mx + b \)

Where \(m\) is the slope and \(b\) is the y-intercept. After utilizing the point-slope form, you often convert the equation to this form to identify these components with ease.
Using our previous equation from the point-slope form, \( y - 1/4 = -34(x - 4) \), we simplify it further:

• First, distribute \(-34\) to obtain: \( y - 1/4 = -34x + 136 \).
• Next, isolate \(y\): \( y = -34x + 136 + 1/4 \).

By converting \(136\) to a fraction \(\left(136 = \frac{544}{4}\right)\), we can easily sum it with \(\frac{1}{4}\) to get:
\( y = -34x + \frac{545}{4} \).
This form clearly shows the line's slope \(-34\) and y-intercept \(\frac{545}{4}\), making it straightforward for further analysis or graphing.