Problem 31 - College Algebra Essentials

Question

If two lines are parallel, describe the relationship between their slopes.

Step-by-Step Solution

Answer

If two lines are parallel, their slopes are equal.

Step 1: Concept of Parallel Lines

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that are both infinite and never intersect. In the context of a Cartesian plane, this means that the lines never cross each other no matter how extended.

Step 2: Slope of a Line

The slope of a line, in mathematics, is a measure of the direction and steepness of a line. It can be computed using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line and \(m\) is the slope.

Step 3: Relationship Between Slopes of Parallel Lines

For two lines to be parallel, their slopes must be equal. This is because slope is a measure of direction and steepness of a line, so two lines with the same slope will have the same direction and will therefore not intersect or cross each other. Hence, if two lines are parallel, their slopes will be equal.