Problem 41 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the given points. \((-3,2)\) and \((4,-7)\)

Step-by-Step Solution

Answer

The slope of the line through the points is \(-\frac{9}{7}\).

Step 1: Identify the Formula for Slope

To find the slope of a line passing through two points, we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Assign Values to the Variables

Label the given points as \((x_1, y_1) = (-3, 2)\) and \((x_2, y_2) = (4, -7)\).

Step 3: Substitute Values into the Formula

Substitute the values into the slope formula: \[ m = \frac{-7 - 2}{4 - (-3)} \].

Step 4: Simplify the Numerical Expression

Calculate the differences in the numerator and denominator: \( -7 - 2 = -9 \) and \( 4 + 3 = 7 \). Therefore, the expression becomes: \[ m = \frac{-9}{7} \].

Step 5: Calculate the Slope

Divide the numerator by the denominator to find the slope: \( m = -\frac{9}{7} \). This is the slope of the line.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to find geometric properties of figures using a coordinate system. It provides a connection between algebra and geometry through graphs and equations. Each point in this system is defined by coordinates represented as tuples. For instance, the point (-3, 2) is described by an x-coordinate of -3 and a y-coordinate of 2.

• Coordinate Plane: It consists of two perpendicular number lines, x-axis and y-axis, intersecting at a point called the origin (0,0).
• Points: Defined as locations on the coordinate plane where these two lines intersect.

This system helps in describing the location of points, shapes, and the relationships between them using algebraic expressions.

Finding Slope

The slope of a line is a measure of its steepness, commonly represented by the letter 'm'. It is calculated as the ratio of the change in y-coordinates to the change in x-coordinates between two distinct points on the line. The slope is key to understanding how one variable in a linear relationship affects the other.

• The slope formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points.
• A positive slope indicates an upward trend and a negative slope reflects a downward trend.

To find the slope between the points (-3, 2) and (4, -7), we substitute into the formula to get \( m = \frac{-9}{7} \). This shows a negative, consistent decline between the two points.

Linear Equations

Linear equations are mathematical expressions used to describe straight lines in algebra and coordinate geometry. They are typically written in the form: \(y = mx + b\) where 'm' represents the slope and 'b' denotes the y-intercept, or where the line crosses the y-axis.

• Slope (m): Tells us how the line inclines up or down depending on its value.
• Intercept (b): Indicates the starting position of the line on the y-axis.

These equations are fundamental for modeling relationships and trends, providing a graphical representation that is easily understood.

Mathematical Problem Solving

Mathematical problem solving involves applying various strategies to find solutions to given problems. It combines critical thinking and analytical skills to work through exercises like finding the slope of a line.

• Understand the Problem: Carefully read what is being asked and determine the known and unknown variables.
• Develop a Plan: Use established formulas and strategies like substituting values into equations.
• Execute the Plan: Perform calculations step by step, being mindful of mathematical operations.
• Look Back: Review the solution to ensure accuracy and completeness.

These steps help in not only achieving correct results but also in building deeper understanding and efficiency in solving similar problems efficiently.