Problem 2 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((2,1)\) and \((3,4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (2,1) and (3,4) is 3, indicating that the line rises.

Step 1: Use the slope formula

The formula for the slope of a line when given two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m = \frac{y_2-y_1}{x_2-x_1}\). Plug the coordinates of the points into the formula. Here, \((x_1,y_1) = (2,1)\) and \((x_2,y_2) = (3,4)\). Thus, \(m = \frac{4-1}{3-2}\)

Step 2: Simplify the slope

Simplify the fraction obtained in step 1 to obtain the slope. In this case, we get \(m = \frac{3}{1}\). Therefore, the slope of the line is 3.

Step 3: Determine line orientation

The slope of the line can determine its orientation. A slope of 0 indicates a horizontal line, an undefined slope (when the denominator of the slope calculation is 0) indicates a vertical line, a positive slope indicates a line that rises (going from left to right), and a negative slope indicates a line that falls (from left to right). Here, as the slope is positive (\(m = 3\)), the line rises.

Key Concepts

Slope Formula

Understanding the slope of a line begins with the slope formula, which mathematically expresses how steep a line is. It's pivotal to grasp this concept as it sets the foundation for much of algebra and calculus. The slope formula is given by
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
where \(m\) represents the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinate pairs of two distinct points on the line. By subtracting the y-coordinates and dividing by the difference in x-coordinates, we find how much the line rises or falls over a horizontal distance. When we say a line 'rises' or 'falls', it does so against the backdrop of a fixed plane. Positive values of \(m\) indicate the line rises as it moves from left to right, while negative values indicate it falls. When the line is perfectly flat, the slope is 0, and thus, the horizontal change has no impact on its height, indicating a horizontal line.

Coordinate Pairs

Coordinate pairs, represented as \((x, y)\), pinpoint the exact position of a point on a two-dimensional plane. Knowing how to interpret and use these pairs is essential in geometry and algebra. In the context of slope, each pair influences the calculation of slope as they mark the starting and ending points of the line segment in question.

For example, to find the slope of the line passing through the points \((2,1)\) and \((3,4)\), we rely on these coordinate pairs to plug into our slope formula. The order in which we subtract these coordinates matters, because reversing the order will change the sign of the slope, but not its absolute value. Hence, it's important to maintain consistency when applying the formula.

Line Orientation

Line orientation refers to the direction in which a line moves across a plane. In a two-dimensional space, lines can rise, fall, or extend horizontally or vertically. The concept of line orientation is directly tied to the sign and value of the slope.

A line with a positive slope, such as \(m = 3\), rises from left to right, indicating an upward trend. Meanwhile, a line with a negative slope will fall or descend as it moves from left to right. Horizontal lines, those parallel to the x-axis, have a slope of 0 and show no rise or fall regardless of the horizontal extension. Vertical lines, which are parallel to the y-axis, have an undefined slope because their change in x is zero—this division by zero is not permissible in mathematics, hence the term 'undefined'. Understanding the orientation of a line is integral to graphing linear equations and to visualizing the relationship between variables represented on the axes.