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Coordinate geometry is a branch of mathematics where we use coordinates or points on a plane to understand geometric concepts. In this context, the points are represented as ordered pairs \(x, y\). Each point on a plane corresponds to a unique position. When we talk about determining the slope of a line, we are essentially looking at the relationship between two such points.

In our original exercise, we have the points A(0,6) and B(4,0). These two points reside on a Cartesian plane, often referred to as the coordinate plane. The plane is comprised of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Point A is at the intersection of the y-axis at y = 6, while Point B is located at x = 4 on the x-axis.

Coordinate geometry helps us visualize lines, curves, and shapes, and allows us to calculate various properties, such as distances, midpoints, and, in this case, the slope of a line.

The slope of a line is a measure of its steepness or incline. In coordinate geometry, it is computed using the slope formula. This essential formula is defined as the change in the y-values divided by the change in the x-values between two points. Mathematically, this is expressed as:

• Slope \(m\) = \(\frac{y_2 - y_1}{x_2 - x_1}\)

where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two points on the line.

In the exercise, the coordinates are A(0,6) and B(4,0), leading to \(x_1 = 0\), \(y_1 = 6\), \(x_2 = 4\), and \(y_2 = 0\). Plugging these values into the formula, we find:

• \(m = \frac{0 - 6}{4 - 0} = \frac{-6}{4}\)

This simple calculation gives us the slope of the line as -1.5, indicating a downward slope from left to right. Understanding the slope formula is crucial, as it not only helps evaluate the steepness of a line but also forms the basis for more complex calculations in coordinate geometry.

Arithmetic simplification plays a key role when working with mathematical formulas, such as the slope formula. Simplification involves reducing an expression to its most basic form. This often involves operations such as addition, subtraction, multiplication, and division.

In our problem, after substituting the given coordinates into the slope formula, we have the expression \(\frac{-6}{4}\). Simplifying this means performing the division to reduce it further, which yields -1.5.

Here are some steps to achieve simplification:

• Perform any basic arithmetic operations, like subtraction or division, step-by-step.
• Reduce fractions to their simplest forms, if applicable.
• Break down complex expressions into simpler components if needed.

Through arithmetic simplification, mathematical problems become easier to understand and solve, allowing you to clearly see the final result. For the slope of the line here, the negative sign in -1.5 also informs us about the direction of the slope, emphasizing its importance as part of the answer.