91Ó°ÊÓ

One of the most common frameworks for writing equations of straight lines is the slope-intercept form. In this form, a line can be expressed as: \[ y = mx + b \]Where:

• \(m\) represents the slope of the line, indicating how steep the line is. A larger absolute value of \(m\) means a steeper line.
• \(b\) is the y-intercept, which denotes where the line crosses the y-axis. This is the point at which the x-value is zero.

The slope-intercept form is popular because it immediately gives you both the slope of the line and where it crosses the y-axis, making visualization and graphing straightforward.

For instance, in the equation presented in the exercise, \(y = \frac{1}{5}x + 6\), \(m\) is \( \frac{1}{5} \) and \(b\) is 6. This tells us the line rises one unit for every five units it moves horizontally and crosses the y-axis at 6.

The point-slope formula is especially useful when you know a point on a line and its slope but not its y-intercept. It is expressed as:\[ y - y_1 = m(x - x_1) \]Where:

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope.

This form is directly derived from the definition of a slope, \((y_2 - y_1)/(x_2 - x_1)\), by multiplying both sides by \(x - x_1\). It's a practical way to quickly plug in a known point and slope to yield the line's equation.

In our case, knowing the line passes through \((2, -3)\) with a perpendicular slope of \(-5\) (found previously), applying the formula gives:\[ y + 3 = -5(x - 2) \]
Point-slope form flexibly shows how the line relates to a specific point, and can be easily manipulated into other forms like the slope-intercept form.

Writing the equation of a line demands understanding its slope and relationship to known points. In this exercise's context, the primary task included finding a line that is perpendicular to another. Here's how it ties together:

• **Slope from Perpendicularity:** The slope of the desired line was calculated using the property of perpendicular lines, which have slopes that multiply to \(-1\). From the original line \( \frac{1}{5} \), the slope of the perpendicular line becomes \(-5\).
• **Utilizing Point and Slope:** With the perpendicular slope known and a point given, we apply the point-slope formula to derive: \( y + 3 = -5(x - 2) \).
• **Converting to Slope-Intercept Form:** Transforming from point-slope to slope-intercept involves distributing and rearranging: \[ y = -5x + 7 \]This step is achieved by solving the equation to isolate \(y\), giving us a clear depiction of the line's characteristics with ease.

Thus, understanding perpendicular slopes and point-slope applications is central to crafting accurate equations in either the point-slope or slope-intercept framework. Whether you start with a point, slope, or interception, each piece aligns to sketch the line's mathematical story.