Problem 2 - Precalculus

Question

Compute the slope of the line passing through the two given points. In Exercise \(3,\) include a sketch with your answers. (a) (-3,0),(4,9) (b) (-1,2),(3,0) (c) \(\left(\frac{1}{2},-\frac{3}{5}\right),\left(\frac{3}{2}, \frac{3}{4}\right)\) (d) \(\left(\frac{17}{3},-\frac{1}{2}\right),\left(-\frac{1}{2}, \frac{17}{3}\right)\)

Step-by-Step Solution

Answer

The slopes are: (a) \( \frac{9}{7} \), (b) \( -\frac{1}{2} \), (c) \( \frac{27}{20} \), (d) -1.

Step 1: Understand the Slope Formula

The slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope.

Step 2: Apply the Slope Formula to (a)

For points (-3, 0) and (4, 9), substitute into the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \): \[ m = \frac{9 - 0}{4 - (-3)} = \frac{9}{4 + 3} = \frac{9}{7} \] Thus, the slope for part (a) is \( \frac{9}{7} \).

Step 3: Apply the Slope Formula to (b)

For points (-1, 2) and (3, 0), substitute into the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \): \[ m = \frac{0 - 2}{3 - (-1)} = \frac{-2}{3 + 1} = \frac{-2}{4} = -\frac{1}{2} \] Hence, the slope for part (b) is \( -\frac{1}{2} \).

Step 4: Apply the Slope Formula to (c)

For points \( \left(\frac{1}{2}, -\frac{3}{5}\right) \) and \( \left(\frac{3}{2}, \frac{3}{4}\right) \), perform the following: \[ m = \frac{\frac{3}{4} - \left(-\frac{3}{5}\right)}{\frac{3}{2} - \frac{1}{2}} = \frac{\frac{3}{4} + \frac{3}{5}}{1} \] \[\frac{3}{4} + \frac{3}{5} = \frac{15}{20} + \frac{12}{20} = \frac{27}{20} \] Thus, the slope for part (c) is \( \frac{27}{20} \).

Step 5: Apply the Slope Formula to (d)

For points \( \left(\frac{17}{3}, -\frac{1}{2}\right) \) and \( \left(-\frac{1}{2}, \frac{17}{3}\right) \), substitute into the slope formula: \[ m = \frac{\frac{17}{3} - \left(-\frac{1}{2}\right)}{-\frac{1}{2} - \frac{17}{3}} \] Convert to common fractions and solve step-by-step: \[\frac{17}{3} + \frac{1}{2} = \frac{34}{6} + \frac{3}{6} = \frac{37}{6} \-\frac{1}{2} - \frac{17}{3} = -\frac{3}{6} - \frac{34}{6} = -\frac{37}{6} \m = \frac{\frac{37}{6}}{-\frac{37}{6}} = -1 \] Thus, the slope for part (d) is -1.

Key Concepts

Slope Formula

The slope formula is a crucial tool in coordinate geometry that helps us understand the steepness and direction of a line. The formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \), calculates the slope \( m \) by taking the difference of the y-coordinates and dividing by the difference of the x-coordinates of two given points. This formula essentially measures how much the line goes "up" or "down" (change in \( y \)) for each step "across" (change in \( x \)).
In simple terms, the slope tells us about the tilt of a line:

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal, indicating no rise.
• If the slope is undefined (division by zero), the line is vertical.

Understanding this concept is fundamental for analyzing linear relationships.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry by using a coordinate plane. This allows us to describe geometric shapes using numerical equations. When working with coordinate geometry, points are defined by pair values \( (x, y) \), which locate their position on the plane.
In the context of slope calculation, coordinate geometry lets us apply algebraic methods to geometric problems. We use the x and y values of two points to determine the line's slope, demonstrating how changes in algebraic expression reflect geometric characteristics on the plane.
This method not only provides a visual understanding of problems but also a precise quantitative analysis, which is essential in many areas of mathematics and applied sciences.

Line Equations

Equations of lines form a core part of understanding linear relationships in coordinate geometry. They provide a way to express all the points on a line mathematically. One of the most common forms of a line equation is the slope-intercept form, \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
The y-intercept is the point where the line crosses the y-axis. By knowing the slope and one point on the line, one can find any other point on the line or determine its equation.
Using the slope-intercept form helps in:

• Quickly identifying the slope and y-intercept.
• Predicting how changes to the equation affect the graph.
• Solving problems involving linear relationships efficiently.

This underlines the importance of the slope not only in finding tilt but also in plotting lines on a graph.

Mathematical Problem-Solving

Mathematical problem-solving is a skill essential for tackling various challenges in mathematics. When dealing with problems like slope calculation, effective problem-solving involves several steps:

• Clearly understanding the problem – knowing what information is given and what is required.
• Analyzing the problem – breaking it down using known formulas, such as the slope formula.
• Applying logical reasoning to interpret results and ensure they fit the given context.

Each of these steps requires practice and an understanding of underlying concepts like slope, coordinate geometry, and line equations.
Patience and persistence are key attributes for successful problem-solving, as they help in finding the right approach and verifying the solution for accuracy. This process of solving an exercise by applying a structured method ensures the development of critical analytical skills.