91Ó°ÊÓ

To find the slope of a line passing through two points, you use the slope formula. This formula is commonly referenced in coordinate geometry and is represented as:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. The slope \(m\) tells you how steep the line is.
In the exercise, we have the points \((3, -5)\) and \((7, 4)\). Substituting into our formula:

• \(m = \frac{4 - (-5)}{7 - 3} = \frac{9}{4}\)

So, the slope is \(\frac{9}{4}\), which means that for every 4 units the line moves horizontally, it rises 9 units.

Once you have calculated the slope, you can use it in the point-slope form to quickly write the equation of a line. The point-slope form is a valuable tool when you know one point on the line and the slope:

• \(y - y_1 = m(x - x_1)\)

This formula uses a chosen point \((x_1, y_1)\) and the slope \(m\) to establish a basis for forming the line's equation.
From our exercise, using point \((3, -5)\) and slope \(\frac{9}{4}\):

• \(y - (-5) = \frac{9}{4}(x - 3)\)

This expresses the relationship between \(x\) and \(y\) using the slope and point information.

Transforming the point-slope form to the slope-intercept form \(y = mx + b\) is straightforward. This form is great for quickly identifying the slope and y-intercept of the equation.
Starting with the point-slope form from earlier:

• \(y + 5 = \frac{9}{4}(x - 3)\)

First, apply the distributive property:

• \(y + 5 = \frac{9}{4}x - \frac{27}{4}\)

Next, solve for \(y\) by subtracting 5 from both sides:

• \(y = \frac{9}{4}x - \frac{27}{4} - 5\)

Convert 5 to a fraction with a denominator of 4:

• \(y = \frac{9}{4}x - \frac{27}{4} - \frac{20}{4}\)

Finally, combine terms:

• \(y = \frac{9}{4}x - \frac{47}{4}\)

The resulting equation shows the line's slope and position where it crosses the y-axis.

Coordinate geometry is a branch of geometry where points are defined using pairs of numbers in a coordinate system. Each point's position is determined by an \((x, y)\) coordinate, making it a powerful tool for understanding spatial relationships between points, lines, and planes.
The essence of coordinate geometry lies in its ability to convert geometric concepts into algebraic formulas. This allows for easier manipulation and application of mathematical operations.
In this exercise, understanding how to use the slope formula and various forms of linear equations—like point-slope and slope-intercept—demonstrates the practical application of coordinate geometry in analyzing and describing lines.