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Understanding the point-line distance formula is crucial when you need to calculate the shortest distance from a point to a line. This formula, derived from the Pythagorean theorem, is used in geometry to measure how far a point is from a line.

The formula is expressed as: \[d = \frac{ |Ax_1 + By_1 + C| }{ \sqrt{A^2 + B^2} }\]
Here, \(d\) represents the distance from the point \((x_1, y_1)\) to the line defined by the equation \(Ax + By + C = 0\). The elements \(A\), \(B\), and \(C\) are coefficients from the equation of the line, and \((x_1, y_1)\) are the coordinates of the given point. The absolute value is used to ensure the distance is non-negative, as distance cannot be negative. The denominator \(\sqrt{A^2 + B^2}\) is the length of the normal vector to the line, which connects the point to the line at a right angle, ensuring the 'shortest' distance is calculated.

The concept of the 'closest point on a line' refers to the point on a line that is nearest to a given external point. This closest point is significant because the line segment connecting the external point to this closest point serves as the perpendicular or normal distance to the line, which is what we measure with the point-line distance formula.

To find the coordinates \((x_2, y_2)\) of the closest point from \((x_1, y_1)\) on a line \(Ax + By + C = 0\), two key equations are used: \[x_2 = x_1 - \frac{A(Ax_{1} + By_{1} + C)}{A^{2} + B^{2}}\]and \[y_2 = y_1 - \frac{B(Ax_{1} + By_{1} + C)}{A^{2} + B^{2}}\]
These equations essentially adjust the given point coordinates by considering how far off the point is from satisfying the original line equation, effectively 'projecting' the point onto the line.

Substituting given values into a formula is a fundamental skill in mathematics used to evaluate expressions and solve equations. Careful substitution is especially important in the context of the point-line distance problem, where precision is needed to obtain the correct distance and closest point coordinates.

To substitute correctly, you should:

• Identify the variables in the formula and the corresponding values from the problem.
• Replace each variable with its value, being mindful of signs and order of operations.
• Simplify the expression following mathematical rules to find your solution.

For example, if you have a point \((x_1, y_1)\) and a line with known coefficients \(A, B,\) and \(C\), you would replace the variables in the distance formula and the equations for the closest point with these known values. Through this process, you would calculate the exact distance from the point to the line and the coordinates of the closest point.