91Ó°ÊÓ

The slope of a line measures its steepness. It is represented by the letter 'm'.
It shows how much the y-coordinate changes for a unit increase in the x-coordinate.
In mathematical terms, the slope between two points \[ (x_1, y_1) \] and \[ (x_2, y_2) \] is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, given points \[ (3, \frac{5}{12}) \] and \[ (5, \frac{9}{10}) \], substitute the coordinates into the formula:
\[ m = \frac{\frac{9}{10} - \frac{5}{12}}{5 - 3} = \frac{\frac{108 - 25}{120}}{2} = \frac{\frac{83}{120}}{2} = \frac{83}{240} \]
This slope tells us how much the y-coordinate changes per unit increase in the x-coordinate.

Slope-intercept form is a common way to write the equation of a line. It is represented as:
\[ y = mx + b \]
Where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis).
Let's convert the point-slope form equation to slope-intercept form using our slope \[ m = \frac{83}{240} \] and the y-intercept 'b'.
Given the point-slope equation:
\[ y - \frac{5}{12} = \frac{83}{240}(x - 3) \]
Simplify to reach slope-intercept form:
\[ y = \frac{83}{240}x - \frac{249}{240} + \frac{25}{60} = \frac{83}{240}x - \frac{249}{240} + \frac{50}{240} = \frac{83}{240}x - \frac{199}{240} \]
Thus, our slope-intercept form equation becomes:
\[ y = \frac{83}{240}x - \frac{199}{240} \]

The point-slope form is another useful way to express the equation of a line. The formula is:
\[ y - y_1 = m(x - x_1) \]
Where \[ (x_1, y_1) \] is a known point on the line and 'm' is the slope.
Using our example points \[ (3, \frac{5}{12}) \] and \[ (5, \frac{9}{10}) \],and the calculated slope \[ m = \frac{83}{240} \], we substitute into the formula:
\[ y - \frac{5}{12} = \frac{83}{240}(x - 3) \]
This form is very useful when you already have a point and need the equation of the line passing through that point.

Coordinate geometry, also known as analytic geometry, studies geometry using a coordinate system. It bridges algebra and geometry through graphs and equations.
It involves plotting points, lines, and curves on a coordinate plane, and using algebraic methods to solve geometric problems.
Important components include:

• Points: Defined by coordinates \[ (x, y) \]. Example: \[ (3, \frac{5}{12}) \]
• Distance: The distance between two points \[ (x_1, y_1) \] and \[ (x_2, y_2) \] is found using the formula:
  \[ \text{Distance} = \frac{\frac{(x_2-x_1)^2 + (y_2 - y_1)^2}} \]
• Midpoint: The midpoint between two points \[ (x_1, y_1) \] and \[ (x_2, y_2) \] is \[ \frac{\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}} \].

The exercise we've solved demonstrates these concepts by identifying points and calculating the slope of a line.
It shows how we use these fundamental ideas to write equations of lines.