91Ó°ÊÓ

Coordinat‪e geometry, also known as analytic geometry, is the study of geometry using a coordinate system. With this, we can describe geometric shapes such as lines and polygons using algebraic equations. In this context, we use ordered pairs \((x, y)\) to represent points on a plane. Each point is defined by its distance from two perpendicular lines called axes—the x-axis and the y-axis.
In our problem, the points given are (\(\frac{3}{4}\), -1) and (\(\frac{-1}{2}\), \(\frac{-1}{2}\)). By plotting these points on a Cartesian plane, we can visualize the line that connects them and better understand their relationship. Coordinate geometry forms the foundation upon which we can calculate other important measurements, like the slope of the line between two points.
The x and y coordinates tell us exactly where the points lie on the grid, making it crucial for slope calculation.

The slope formula is a fundamental concept in coordinate geometry and is used to measure the steepness, incline, or grade of a line. It is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \(m\) is the slope, \(x_1, y_1 \) are the coordinates of the first point, and \(x_2, y_2\) are the coordinates of the second point. That means it defines how much the y-coordinate changes for a unit change in the x-coordinate.
In our problem:

• \(x_1 = \frac{3}{4}\)
• \(y_1 = -1\)
• \(x_2 = -\frac{1}{2}\)
• \(y_2 = -\frac{1}{2}\)

We substitute these values into the slope formula to find the gradient of the line.

The difference of coordinates is another vital concept, which involves subtracting the corresponding coordinates of the two points.
First, calculate the difference in y-coordinates (\(y_2 - y_1\)):
\[ y_2 - y_1 = -\frac{1}{2} - (-1) = -\frac{1}{2} + 1 = \frac{1}{2} \]
Next, calculate the difference in x-coordinates (\(x_2 - x_1\)):
\[ x_2 - x_1 = -\frac{1}{2} - \frac{3}{4} = -\frac{1}{2} - \frac{3}{4} = -\frac{5}{4} \]
These differences are then used in the slope formula to find the slope between the two points. It's important to carefully handle signs and fractions in these calculations.

To find the slope, we often deal with fractions, so understanding fraction operations is crucial.

• Addition/Subtraction: To add or subtract fractions, make sure the denominators are the same. Otherwise, find a common denominator and adjust the fractions accordingly.
• Multiplication: Multiply the numerators together and the denominators together.
• Division: Multiply by the reciprocal of the divisor.

In our problem, after finding the differences in coordinates, we get:
\[ m = \frac{\frac{1}{2}}{-\frac{5}{4}} \]
To divide fractions, we multiply by the reciprocal of the second fraction:
\[ m = \frac{1}{2} \times \frac{4}{-5} = \frac{4}{-10} = -\frac{2}{5} \]
Through this step-by-step process, you can confidently handle fractions and accurately determine the slope.