The shortest distance from the line \(3 x-4 y=3\) to the point \((6,8)\) is a) \(4.8\) b) \(4.2\) c) \(3.8\) d) \(3.4\)

Understanding the equation of a line is critical when dealing with geometric problems. A line equation can be expressed in various forms, but one of the most common is the standard form, written as \( Ax + By = C \). This linear equation represents a line on the Cartesian coordinate system, where \( A \), \( B \), and \( C \) are constants that affect the slope and the position of the line on the plane.

For the given line equation \(3x - 4y = 3\), the constants are \( A = 3 \), \( B = -4 \), and \( C = 3 \). These constants can be used to determine several properties of the line, such as its slope and y-intercept, which are essential in understanding the overall geometry of the line. However, in our context, they are directly used to find the distance from a specific point to the line.

It's vital to recognize the standard form of a line, as this will be the starting point for many geometrical problem-solving exercises, such as finding intersections, angles between lines, and as in our case, distances from points to a line.

In a Cartesian coordinate system, the location of any point is specified by its coordinates, shown as an ordered pair \( (x, y) \). The first value, \( x \), is the horizontal position relative to the origin, and the second value, \( y \), is the vertical position.

For the problem at hand, the point given is \( (6, 8) \). These coordinates are integral to calculating the distance from this point to a line. This ordered pair provides the specific values that will be substituted into algebraic formulas or methods designed to assess the point's relationship to the line, such as the perpendicular distance from the point to the line.

Plotting Points

Plotting points based on their coordinates lays the groundwork for visualizing the position of the point relative to the line. Visual aids, although not always necessary, can significantly help in understanding the problem's geometry. Being comfortable with plotting and interpreting point coordinates is essential for navigating through more complex problems in analytic geometry.

Numerical problem solving includes using arithmetic operations, algebraic manipulations, and formula applications to find numerical solutions to mathematical problems. Each step must be carried out with precision to avoid errors that can lead to incorrect conclusions. In geometric contexts, numerical problem solving often involves substituting values into formulas and simplifying.

In our scenario, after recognizing the formula for the distance from a point to a line, numerical problem solving comes into play. The accuracy of each mathematical operation - from ensuring the correct sign while substituting values to performing the arithmetic operations and square root calculations - is crucial.

Double-Checking Calculations

It is essential to always double-check calculations especially when the final answer does not align with expected outcomes, as seen in our exercise problem. In our case, after performing the necessary calculations, if we identify that the provided answer choices do not correspond to our solution, we need to reassess each step to ensure no mistakes have been made. Detailed understanding and diligence in numerical procedures are key to achieving the right solutions.