In Exercises 13-16, write an equation of the line passing...

In Exercises $13-16,$ write an equation of the line passing through point $\mathrm{P}$ that is parallel to the given line. Graph the equations of the lines to check that they are parallel. (See Example $3 . )$

$$\mathrm{P}(3,8), \mathrm{y}=\frac{1}{5}(\mathrm{x} 14)$$

Key Concepts

Slope

The slope of a line is a measure of its steepness, represented as the ratio of the vertical change between two points to the horizontal change. It is a fundamental concept in analytic geometry used to determine the direction of a line. In the context of parallel lines, understanding the slope is crucial because parallel lines have the same slope.

Parallel Lines

Parallel lines are lines in a plane that never intersect, regardless of how far they are extended. This geometric property is directly linked to their slopes; if two lines are parallel, they must have identical slopes. This concept is essential when writing the equation of a line that is required to be parallel to a given line.

Point-Slope Form

The point-slope form is a method to write the equation of a line when a point on the line and its slope are known. It is expressed in the form y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This approach is particularly useful when determining the equation of a line parallel to a given line, as the slope is already known from the given line.

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the equation on a coordinate plane and drawing a line through these points. This visual representation helps to verify properties such as parallelism or intersection. Graphing is a practical tool that reinforces the correctness of the algebraic solution and aids in understanding the spatial relationships between lines.

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