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The equation of a line is fundamental in geometry and plays a critical role in finding the distance from a point to a line. It is usually expressed in the form \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are constants that uniquely define the line's slope and position on the coordinate plane. When a line equation is given, you can use these constants to understand the orientation and steepness of the line.

For example, in the exercise \(4x - 3y = 0\), \(A = 4\), \(B = -3\), and \(C = 0\). Here, the line has a slope of \(\frac{4}{3}\) and passes through the origin since \(C\) is zero. Understanding the line equation is the first step to calculating the distance from a point to this line.

Moving forward, the distance formula is crucial for measuring the shortest distance between a point and a line. The formula is derived from the Pythagorean theorem and is represented as:

\[d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\]

where \(d\) is the distance, \(x_1\) and \(y_1\) are the coordinates of the point, and \(A\), \(B\), and \(C\) are parameters from the line equation. The formula's numerator consists of the absolute value of the expression found by plugging the point's coordinates into the line equation. The denominator is the square root of the sum of squares of \(A\) and \(B\), which relates to the line's slope. This formula gives a precise, direct distance regardless of the position of the point relative to the line.

Finally, the point-line distance calculation involves applying the distance formula to the specific case at hand. After identifying the constants \(A\), \(B\), and \(C\) from the line equation and the coordinates \(x_1\) and \(y_1\) from the given point, substitute these values into the distance formula and simplify as needed.

In the given exercise, for the point (1,2) and the line equation \(4x - 3y = 0\), the computation is straightforward: replace \(A\), \(B\), \(C\), \(x_1\), and \(y_1\) into the formula to get \[d = \frac{|4(1) - 3(2)|}{\sqrt{4^2 + ( -3)^2}}\] which simplifies to \[d = \frac{2}{\sqrt{25}}\], resulting in a distance of \(\frac{2}{5}\). This step-by-step approach helps ensure that students clearly understand how the variables and constants relate and interact within the formula to yield the result.