Problem 7 - College Algebra

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. $$(-2,4)\( and \)(-1,-1)$$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (-2,4) and (-1,-1) is -5. The line through these points falls downwards.

Step 1: Substitute in the formula

First, substitute the given point values into the slope formula. Here, \((x_1, y_1) = (-2, 4)\) and \((x_2, y_2) = (-1, -1)\). Hence, the slope, \(m\), can be calculated as follows: \(m = (y_2 - y_1) / (x_2 - x_1) = ((-1) - 4) / ((-1) - (-2)).

Step 2: Simplify the slope

Calculate the differences in the numerator and the denominator and then compute the division to get the slope. Here, \(m = (-1 - 4) / -1 + 2 = -5 / 1 = -5\).

Step 3: Indicate line direction

Based on the sign of the slope, we can conclude the direction of the line. Here, the slope is -5, which is a negative number. Hence, the line through the points falls since negative slope indicates downward direction.

Key Concepts

Slope Formula

Understanding how to calculate the slope of a line is fundamental in algebra and geometry. The slope is a measure of how steep a line is on a graph. The slope formula, which is \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), is used to find this rate of change. The variables \( y_2 \) and \( y_1 \) represent the y-coordinates of two different points on the line, while \( x_2 \) and \( x_1 \) represent the x-coordinates. By inserting the respective coordinate values into this formula, as shown in our original exercise, we find the slope by subtracting the y-coordinates and dividing by the difference of the x-coordinates.

It is important to follow the correct order of points when substituting them into the formula. The reason is to maintain consistency and avoid getting a slope with an incorrect sign. Consistent use of this formula helps to reveal important characteristics of the line, like its direction, which we discuss further below.

Coordinate Points

Coordinate points are the specific locations of points on a two-dimensional graph. Each point is represented by a pair of numbers, \( (x, y) \), where \( x \) indicates the horizontal distance from the origin, and \( y \) indicates the vertical distance from the origin. In the context of finding slopes, these points allow us to understand the relationship between points on a line.

For example, in the problem provided, the coordinates \( (-2, 4) \) and \( (-1, -1) \) indicate the placement of two points on a Cartesian plane. When we calculate the differences between the y-coordinates and the x-coordinates of these points, we essentially measure the horizontal and vertical distances the line travels between these two points, which directly informs us about the steepness and direction of the line, leading to understanding the slope. As such, the accuracy in plotting and reading coordinate points is essential for successfully applying the slope formula.

Line Direction

The direction of a line is indicated by the sign of its slope - whether the slope is positive, negative, zero, or undefined. In our exercise, we found a negative slope, implying that the line falls or has a downward trajectory as we move from left to right across the graph.

A positive slope means the line rises; it goes up as we move from left to right. A slope of zero, which occurs when \( y_2 - y_1 = 0 \), indicates a horizontal line. Conversely, an undefined slope, resulting from a division by zero in the case where \( x_2 - x_1 = 0 \), indicates a vertical line. Understanding the direction of a line is crucial as it gives us visual insight into the relationship between variables represented on a graph - it can indicate, for instance, whether an increase in one variable results in an increase or decrease in another.