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Understanding coordinates of a point is foundational to analytic geometry. Every point on a two-dimensional plane is determined by two numbers, known as its coordinates. Typically denoted as (x, y), these two numbers represent the point's position relative to two perpendicular lines called axes.

For example, the coordinates (1, 4) mean that from the origin (0, 0), where the x-axis and y-axis intersect, you move one unit right and four units up to locate the point. The origin serves as the reference for all the coordinates on the plane. In the context of calculating distances, knowing the exact location of the point is crucial, as it is one of the components used in the distance formula.

The line equation in algebra is a way to describe a straight line through a mathematical formula. There are various forms to represent it, but one of the most commonly used is the slope-intercept form, which is written as y = mx + b. In this equation, 'm' represents the slope of the line, and 'b' gives the y-intercept, which is where the line crosses the y-axis.

For the equation of the line provided in the exercise, y = 4x + 2, '4' is the slope, and '2' is the y-intercept. It means that for every one unit you move right along the x-axis, the line goes up by four units. When you need to find the distance from this line to a specific point, understanding the line's properties and being able to express it in different forms, such as the general form Ax + By + C = 0, is essential for using the distance formula.

The distance formula is a valuable tool in geometry, allowing us to calculate the shortest distance between a point and a line. As seen in the provided example, the formula is \(d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}\), where A, B, and C are coefficients from the line equation Ax + By + C = 0, and \(x_0, y_0\) are the coordinates of the point in question.

The formula uses the absolute value to ensure the distance is always a positive measure since distance cannot be negative. The denominator \(\sqrt{A^2+B^2}\) normalizes this distance, taking into account the slope of the line. Sometimes, the distance needs to be rationalized to avoid having a radical in the denominator, making the expression tidier as in the final answer \(\frac{2\sqrt{17}}{17}\). Understanding how to manipulate and apply the distance formula is pivotal in solving many geometric problems.