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The slope-intercept form of a line is a handy way to express the equation of a straight line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis.

Using slope-intercept form is great because it gives you a quick snapshot of the line's behavior. The slope \( m \) tells us how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. The y-intercept \( b \) offers another crucial piece by showing where the line hits the y-axis.

For example, in the step-by-step solution of our exercise, we turned the provided equation into slope-intercept form to find the slope. This is important for determining the slope of the perpendicular line, a crucial step in writing a new line's equation.

An equation of a line is a mathematical description of a straight line on a graph. It tells us everything about the line, like its slope and position. There are different ways to write the equation of a line, such as the slope-intercept form and the point-slope form.

The importance of a line equation lies in its ability to pinpoint how a line behaves on a coordinate plane. It allows us to understand the line's direction and where it lies relative to the x and y axes. For solving geometric problems, this becomes invaluable.

In our exercise, finding the new line's equation was crucial because it indicated how a perpendicular line behaves compared to the original line. Understanding the equation enables you to draw or analyze the line without needing to directly visualize it.

The point-slope form is another way to write the equation of a line, especially useful when you know a point on the line and its slope. The formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the known point, and \( m \) is the slope of the line.

This form is particularly helpful in problems like the exercise where you need to find the equation of a line passing through a certain point with a given slope. Once you have a slope and a specific point, the rest is just plugging in values!

In the exercise, we started with a known point and a newly calculated slope for the perpendicular line. Using the point-slope form made it easy to establish the equation of the new line. Ultimately, this form provided a simple method to navigate from known slopes and points to the line's full equation.