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Coordinate Geometry is a branch of geometry where we use a coordinate system to define the positions of points. In two-dimensional coordinate geometry, we usually use the Cartesian plane, which consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are represented by coordinates \(x, y\).

In our problem, the equation \(4x - 3y + 15 = 0\) represents a line on the Cartesian plane. Finding the distance from the origin, \( (0,0) \), to this line requires understanding the geometry of the situation.

Key concepts of Coordinate Geometry include:

• Points: Defined by coordinates (x, y).
• Lines: Represented by linear equations, like in our equation \(4x - 3y + 15 = 0\).
• Distance: Calculated using a specific formula when measuring between a point and a line.

These foundational ideas help in solving problems related to shapes, sizes, and the relative position of figures on a plane.

Linear Equations play a vital role in solving various problems in mathematics, including our exercise. A linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. It is typically written in the form \(Ax + By + C = 0\). In this format:

• A, B, C: are constants.
• (x, y): are variables representing coordinates.

Linear equations can help us analyze the relationship between variables, allowing us to identify the slope and position of a line on the graph.

In our exercise problem, by identifying the constants as \(A=4\), \(B=-3\), and \(C=15\), we can substitute these into the formula for computing the distance from a point to a line. This step simplifies the problem, highlighting how linear equations convert geometric problems into solvable algebraic expressions.

Algebraic Expressions are combinations of numbers, variables, and mathematical operations. They form the building blocks of equations and inequalities, crucial for solving problems like the one in our exercise.

The equation \(4x - 3y + 15 = 0\) can be seen as an algebraic expression where the variables are manipulated to understand the properties and solutions of the problem.

Key points about algebraic expressions include:

• Constants and Variables: Constants are fixed values, while variables represent unknown quantities.
• Operations: Operations such as addition, subtraction, multiplication, and division combine these elements.

In solving this exercise, we used algebraic manipulation to derive the distance formula from the expression given. This highlights how algebraic expressions transform theoretical problems into practical solutions by organizing and simplifying complex mathematical ideas.