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Finding the slope of a line is a key concept in coordinate geometry and linear equations. The slope indicates how steep a line is and the direction it moves. We can find the slope using the formula: \( \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \). This formula computes the ratio of the vertical change (rise) to the horizontal change (run) between two points.

In the example given, we have points \( (x_1, y_1) = (2, -2) \) and \( (x_2, y_2) = (6, 5) \). Substituting these values into our formula: \( \frac{5 - (-2)}{6 - 2} = \frac{7}{4} \). The slope is \( \frac{7}{4} \), which means for every 4 units of horizontal movement, the line goes up 7 units.

Remember that:

• A positive slope means the line rises as you move from left to right.
• A negative slope means the line falls as you move from left to right.
• A slope of zero means the line is horizontal.
• An undefined slope (division by zero) means the line is vertical.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This form of geometry allows for the precise representation and analysis of geometrical shapes and lines. The key components are:

• Points: Defined by coordinates \( (x, y) \) in a 2D space.
• Lines: Can be defined by the slope and a point, using linear equations.
• Distance: The Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( \text{d} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

For instance, to find the distance between points \( (2, -2) \) and \( (6, 5) \), we use the formula:

\( \text{d} = \sqrt{(6 - 2)^2 + (5 - (-2))^2} = \sqrt{16 + 49} = \sqrt{65} \)

Coordinate geometry is essential for understanding the relationship and properties of geometric shapes in an algebraic context.

Linear equations represent the relationship between variables that results in a straight line when graphed. The most common form is the slope-intercept form:

\( y = mx + b \)

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

Using our previous example, with a slope of \( \frac{7}{4} \), if we know a point \( (x_1, y_1) = (2, -2) \), we can find the equation of the line:

\( y - y_1 = m(x - x_1) \)

Substitute the known values:

\( y + 2 = \frac{7}{4}(x - 2) \)

Distribute the slope:

\( y + 2 = \frac{7}{4}x - \frac{7}{2} \)

Rearrange to slope-intercept form:

\( y = \frac{7}{4}x - \frac{7}{2} - 2 \)

Simplify the constants:

\( y = \frac{7}{4}x - \frac{11}{2} \)

Linear equations are fundamental in coordinate geometry, helping us understand how changes in one variable affect another.