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The slope-intercept form is a straightforward way to express the equation of a line. It is represented by the equation \(y = mx + b\). In this formula:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

To find the equation of a line, you need these two components. Let's break them down:First, you calculate the slope \(m\) using two points on the line. For instance, with points \((3,6)\) and \((0,-3)\), you find the slope as \(m = \frac{(-3-6)}{(0-3)} = 3\).

Next, determine the y-intercept \(b\) by solving \(b = y - mx\). Using one of the given points, such as \((3,6)\), you substitute: \(b = 6 - 3 \times 3 = -3\).

Finally, you plug \(m\) and \(b\) into the slope-intercept formula, resulting in \(y = 3x - 3\), which is the equation of the line.

The point-slope formula is another method to write the equation of a line, often useful when you know a point on the line and its slope. The formula is expressed as:\[ y - y_1 = m(x - x_1) \]In this equation:

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope.

This form is particularly handy if you need to write the equation quickly and build from one known point and the slope. Using a point such as \((3,6)\) and slope \(m = 3\):\[ y - 6 = 3(x - 3) \]
This simplifies to:\[ y = 3x - 3 \]
You've arrived at the same line equation, reaffirming the consistency between point-slope and slope-intercept forms.

Coordinate geometry, or analytic geometry, deals with the representation of geometric figures using coordinates. It's a key part of understanding the equations of lines, circles, and other shapes in the coordinate plane.

Understanding the line equation in coordinate geometry begins with plotting points on the Cartesian plane, which has a horizontal axis (x-axis) and a vertical axis (y-axis).

• Every point is in the form \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.

In our case, the line passing through \((3,6)\) and \((0,-3)\) indicates that we need to understand both the slope and positioning on this plane.

Points and their arrangements help us find distances, midpoints, and slopes, facilitating just how we derive the line equation through calculations like those shown in the exercise. Comprehending these connections makes coordinate geometry indispensable for tackling more complex scenarios like curves and transformations.