91Ó°ÊÓ

The coordinate plane is a two-dimensional surface where points are plotted using a pair of numerical coordinates. These coordinates are written in the form \((x, y)\), where \(x\) corresponds to the horizontal position and \(y\) to the vertical position.
The coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is known as the origin, represented as \((0, 0)\).
Using this plane, we can easily visualize equations and geometrical shapes like lines, as well as identify exact positions of specific points or intersections.

The x-intercept of a line is the point where the line crosses the x-axis. At this intersection, the y-coordinate is always zero because the point lies directly on the x-axis.
For example, in the task’s scenario, the x-intercept is given as 3. This means that the line crosses the x-axis at the point \((3, 0)\). Understanding the x-intercept is crucial for graphing, as it provides a specific anchor point that helps define the line's direction on the coordinate plane. It represents an important feature of the line since it highlights where the line ceases to rise or fall within the vertical axis.

The y-intercept is similarly significant, as it indicates the point where the line crosses the y-axis. Here, the x-coordinate is zero because the point is located on the y-axis itself.
In our exercise, the y-intercept is 4, so the line crosses the y-axis at the point \((0, 4)\). Like the x-intercept, the y-intercept serves as another foundational anchor point on the graph. By identifying both intercepts, you provide two essential locations to accurately draw the line on the coordinate plane, giving complete insight into its positioning and slope.

The slope of a line indicates its steepness and direction. A negative slope means that as you move along the line from left to right, the line falls or goes downward. This shows a decrease in the y-values since the line is descending.
In mathematical terms, the slope \(m\) can be calculated by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), derived from two points \((x_1, y_1)\) and \((x_2, y_2)\).
In the problem, where points are \((0, 4)\) and \((3, 0)\), substituting the values gives us\[m = \frac{0 - 4}{3 - 0} = -\frac{4}{3} \].
Hence, this negative slope of \(-\frac{4}{3}\) confirms that the line slopes downward, aligning with the visual observation made from the drawing on the coordinate plane.