Problem 25 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$(1,7) \text { and }(-4,2)$$

Step-by-Step Solution

Answer

The slope of the line containing the points \((1,7)\) and \((-4,2)\) can be found using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given values, we get \(m = \frac{2 - 7}{-4 - 1} = \frac{-5}{-5} = 1\). Thus, the slope of the line is \(1\).

Step 1: Identify the coordinates of the given points

We are given the two points \((1,7)\) and \((-4,2)\). Thus, we have: - \(x_1 = 1\) - \(y_1 = 7\) - \(x_2 = -4\) - \(y_2 = 2\)

Step 2: Apply the slope formula

We know the slope formula is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Substitute the values of \(x_1, y_1, x_2,\) and \(y_2\) into the formula: \[m = \frac{2 - 7}{-4 - 1}\]

Step 3: Simplify the expression

Now, work through the subtraction in the numerator and the denominator: \[m = \frac{-5}{-5}\] Finally, divide the numerator by the denominator: \[m = 1\]

Step 4: Interpret the result

The slope of the line containing the points \((1,7)\) and \((-4,2)\) is \(1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. This branch of mathematics allows us to find relationships between figures using algebraic equations and coordinates.

In everyday usage, points are plotted on a grid with horizontal and vertical axes, often labeled as the "x-axis" and "y-axis." Each point is represented by an ordered pair \( (x, y) \), showing its position on the coordinate plane.

For example, the points \( (1,7) \) and \( (-4,2) \) lie in a two-dimensional space, which enables us to draw the line that connects them. By using these coordinates, we can analyze and describe the line in precise mathematical terms.

If you picture this, you can imagine drawing horizontal and vertical lines from each point to determine how far they are from one another in the x-direction and y-direction. Using the coordinate system makes it simple to visualize and calculate properties of geometric figures.

Linear Equations

Linear equations represent straight lines and are fundamental in understanding relationships between variables. The equation for a line can usually be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In our exercise, we determined the slope \( m = 1 \) for the line passing through the points \( (1,7) \) and \( (-4,2) \). The slope tells you how steep the line is; in this case, a slope of one means the line rises one unit for every unit it moves horizontally.

Linear equations allow us to make predictions and understand linear relationships. By knowing the slope and a point on the line, you can construct the complete equation for that line.

• Slope \( m \) is crucial as it tells us how y changes with x.
• The y-intercept \( b \) is where the line crosses the y-axis, giving a starting point for graphing.

Algebra 1

Algebra 1 is an introductory course that focuses on using letters and symbols to represent numbers and express mathematical relationships. This foundational subject introduces essential concepts like variables, equations, and functions.

During our study of slopes, we use algebraic symbols to simplify expressions and solve equations. For example, in finding the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \), we plug in the coordinates and simplify:

1. Subtract the y-values: \( y_2 - y_1 = 2 - 7 = -5 \)
2. Subtract the x-values: \( x_2 - x_1 = -4 - 1 = -5 \)
3. Simplify the fraction: \( \frac{-5}{-5} = 1 \)

Algebraic skills allow us to approach complex problems in a structured way. These skills are essential tools in solving real-world problems, modeling situations, and understanding the patterns of numbers.