Problem 49 - Elementary Algebra

Question

Find the slope, if it exists, of the line through the given pairs of points. $$ (1,-4), \quad(3,3) $$

Step-by-Step Solution

Answer

Based on the step by step solution, determine the slope of the line that passes through the points (1, -4) and (3, 3). Answer: The slope is $\frac{7}{2}$.

Step 1: Identify the given points

We are given the points \((1, -4)\) and \((3, 3)\). These points will be used in the slope formula to find the slope of the line through them.

Step 2: Write down the slope formula

The slope formula is given as follows: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ We will plug in the given coordinates into the formula to find the slope.

Step 3: Plug in the coordinates

Using the given coordinates, we plug in the values into the slope formula as follows: $$ m = \frac{3 - (-4)}{3 - 1} $$

Step 4: Calculate the slope

Now we simplify the expression: $$ m = \frac{3 + 4}{3 - 1} \\ m = \frac{7}{2} $$ The slope of the line through the given pairs of points is \(\frac{7}{2}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. It brings together algebra and geometry to describe geometric figures and their properties in terms of a coordinate plane. In the context of our problem, this involves understanding how to locate points in a two-dimensional space. The points are represented as pairs of numbers, such as \( (1, -4) \) and \( (3, 3) \), which provide the x and y coordinates of each point.

To visualize, imagine a grid with horizontal and vertical lines, where each point on the plane has a unique position determined by these coordinates.

• The horizontal line represents the x-axis.
• The vertical line represents the y-axis.
• Each point on the plane can be described with an ordered pair \( (x, y) \).

These points form the basis of defining shapes, determining distances, and calculating slopes, which describe the steepness or incline of a line in this plane.

Linear Equations

Linear equations are expressions that represent a straight line when graphed on a coordinate plane. A fundamental component of linear equations is their slope, which depicts how quickly one variable changes with respect to another.

A standard form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this context, the slope, denoted by \( m \), is crucial as it influences the angle and direction of the line across the plane.

• A positive slope means the line rises as it moves from the left to the right.
• A negative slope indicates it falls.
• A zero slope results in a horizontal line, indicating no rise or fall.
• An undefined slope is a vertical line, moving neither left nor right.

Knowing how to derive the slope between two points as done in the exercise helps in graphing and understanding the behavior of linear equations.

Mathematical Formula

The mathematical formula for calculating the slope of a line is vital in coordinate geometry and appears often in tasks requiring analysis of linear relationships. This formula allows us to quantify the change between two points on a plane. Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated using:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• You subtract the y-coordinates to find the change in vertical distance.
• You subtract the x-coordinates to find the horizontal change.
• The resulting fraction \( \frac{\Delta y}{\Delta x} \) represents the slope.

In our problem, applying the formula to points \( (1, -4) \) and \( (3, 3) \) yields \( m = \frac{7}{2} \), indicating that for every 2 units moved horizontally, the line climbs 7 units vertically. This fundamental mathematical tool helps students bridge algebra and geometry, providing clarity in understanding the slope's geometric and algebraic implications.