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The distance between a point and a line in a two-dimensional space can be calculated using a specific formula. This is crucial in coordinate geometry and helps us find out how far a given point is located from a line. The distance formula is given by:

• \[ d = \frac{{| ax_1 + by_1 + c |}}{{\sqrt{{a^2 + b^2}}}} \]

Where:

• \( a \), \( b \), and \( c \) are the coefficients from the line equation \( ax + by + c = 0 \)
• \( (x_1, y_1) \) are the coordinates of the point

This formula calculates the perpendicular distance, which is the shortest distance from a point to a line. By substituting the coefficients and the point's coordinates into this formula, you can determine the required distance easily.

Coordinate geometry, also referred to as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric properties of figures and objects. It allows us to understand spatial relationships through a coordinate system.
In a two-dimensional coordinate system, each point is represented as an ordered pair \((x, y)\). The equations of straight lines, such as the one used in our example \(2x - y = 4\), can describe lines in this plane.

• The equation of any line can be rearranged into the form \(ax + by + c = 0\).
• This makes it easier to manipulate and work with lines when finding solutions like distances or intersections.

Understanding these concepts helps us solve various geometric problems using algebraic methods.

Linear equations are algebraic expressions involving variables raised only to the first power. This simplicity makes them a fundamental concept in both algebra and geometry. In our problem, the line is expressed as a linear equation:

• \(2x - y = 4\)

To use this equation in calculations, it might often be helpful to express it in standard form \(ax + by + c = 0\).
The coefficients \(a\), \(b\), and the constant \(c\) make it straightforward to apply formulae like the distance formula. Linear equations can represent lines on a graph, with each unique equation corresponding to a distinct line.
In problems involving distances or intersections, manipulating the linear equation correctly is key to finding the right solutions.