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In coordinate geometry, two lines are considered parallel if they have the same slope but different y-intercepts. This means they will never intersect, no matter how far they extend. Parallel lines can be identified easily by comparing their slopes.

For example, the equation of the line given in the exercise is parallel to the x-axis. Lines parallel to the x-axis have a unique property: their slope is always zero. Thus, any line parallel to the x-axis will be of the form y = constant.

Remember! Parallel lines have the same slope. For vertical lines parallel to the y-axis, the equation will be in the form of x = constant. Knowing these patterns makes it easier to identify and work with parallel lines.

The slope of a line is a measure of its steepness. It is commonly denoted by the letter 'm' in the equation of a line. The slope is calculated as the ratio of the vertical change to the horizontal change between two points on the line: \ \( m = \frac{{\text{{rise}}}}{{\text{{run}}}} \ \).

In the case of lines parallel to the x-axis, their slope is always zero. This is because there is no vertical change as you move along the line, resulting in \ ( \text{{rise}} = 0 \ ), and thus, the slope \ ( m = 0 \ ).

Understanding the slope helps in determining the angle and direction of a line. When you know the slope, you can easily write the equation of the line using the slope-intercept form: y = mx + b, where 'b' represents the y-intercept.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows for the representation of geometric shapes algebraically and provides a link between algebra and geometry.

The coordinates of a point are represented as (x, y), where 'x' is the horizontal distance from the origin, and 'y' is the vertical distance. This system makes it possible to write equations representing lines, curves, and other geometric shapes.

In the given exercise, we use the coordinate system to determine the equation of a line parallel to the x-axis that passes through the point (3, 5). Knowing the characteristics of lines parallel to the x-axis, we easily find the equation to be y = 5.

By understanding coordinate geometry, you can solve many problems related to distances, midpoints, gradients, and intersections, all of which are crucial in both algebra and geometry.