Problem 7 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) 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. $$(-2,4) \text { and }(-1,-1)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-2,4) and (-1,-1) is -5. The line is a falling line as it has a negative slope.

Step 1: Recall the formula

Recall the formula for finding the slope of a line given two points \((x1, y1)\) and \((x2, y2)\) is \[m=\frac{y2-y1}{x2-x1}\]

Step 2: Identify the given points

We are given the points \((-2,4)\) and \((-1,-1)\) which can be labeled as \((x1, y1)\) and \((x2, y2)\) respectively.

Step 3: Plug the points into the slope formula

Plugging these values into the slope formula gives us \[m=\frac{-1-4}{-1-(-2)}\]

Step 4: Simplify the expression

Simplifying this gives us \[m=\frac{-5}{1}\]

Step 5: Analyze the slope

So the slope of the line is -5. Since the slope is negative, this indicates that the line falls.

Key Concepts

Line Through Points

Finding the slope of a line that passes through two points is a fundamental concept in understanding the geometry of lines. Whenever you have two distinct points, like \((-2,4)\) and \((-1,-1)\), these form the basis for calculating the line’s slope and derive its behavior.In this context:

• A line through points determines the steepness and direction of the line. By defining two points, you can calculate how much the line inclines or declines.
• To compute the slope, you apply the slope formula \(m = \frac{y_2-y_1}{x_2-x_1}\).
• This formula measures the "vertical change" (rise) relative to the "horizontal change" (run) between these points.

Determining the line through two points is the foundational step for assessing characteristics like its slope, direction, and overall graphing on a coordinate plane.

Negative Slope

A negative slope is indicative of a specific direction and movement of a line when graphed. In simple terms, a line with a negative slope falls or declines from left to right.When you calculated the slope for points \((-2,4)\) and \((-1,-1)\), you found it to be \(m = -5\).

• A negative slope means that for each unit move to the right along the x-axis, the line moves downward along the y-axis.
• The magnitude of the slope tells us the steepness of the line. A larger absolute value indicates a steeper downfall.
• Here, with a slope of \(-5\), the line drops 5 units down for every 1 unit it moves right.

Recognizing a negative slope gives you immediate insight into how the line behaves graphically.

Rise or Fall of a Line

Understanding whether a line rises or falls is important for comprehending its behavior visually and mathematically. This rise or fall relates to the sign and value of the slope.For the given line with slope \(m = -5\):

• The negative sign of the slope directly tells you that the line falls. This falling nature is consistent across any line with a negative slope.
• A falling line moves downward as it travels from the left to the right of the graph.
• The rate of this fall is determined by the slope's value. Since \(-5\) is relatively steep, the line will fall sharply.

Recognizing whether a line rises or falls provides clarity on how changes along the x-axis affect the position along the y-axis.