Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-28, find the slope of the line containing the given two points. \((-6,0)\) and \((0,6)\)

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Understanding the Slope Formula

The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This equation is used to determine the change in \(y\) values relative to the change in \(x\) values between the two points.

Step 2: Identify the Points

Identify the coordinates of the given points: \((-6,0)\) as \((x_1, y_1)\) and \((0, 6)\) as \((x_2, y_2)\).

Step 3: Substituting the Coordinates

Substitute the coordinates into the slope formula: \( m = \frac{6 - 0}{0 - (-6)} \).

Step 4: Simplifying the Expression

Simplify the expression to calculate the slope: \( m = \frac{6}{0 + 6} = \frac{6}{6} = 1 \).

Step 5: Conclusion

The slope of the line through the points \((-6,0)\) and \((0,6)\) is \(1\). This indicates that for every unit change in \(x\), \(y\) increases by \(1\).

Key Concepts

Linear Equations

Linear equations are a fundamental part of mathematics, especially when exploring lines on a graph. These equations are written in the form of \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
One of the key purposes of linear equations is to model relationships that are proportional, meaning they have a constant rate of change. This rate of change is precisely what the slope measures. To calculate the slope when given two points, like \((-6,0)\) and \((0,6)\), we use the slope formula:

• The difference in the y-values, \( y_2 - y_1 \), shows how much the line rises or falls vertically between two points.
• The difference in the x-values, \( x_2 - x_1 \), indicates the horizontal change.

By dividing these differences, we find the slope, illustrating how steep the line is.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry using the coordinate plane.This allows us to describe the location of points using ordered pairs and to understand spatial relationships through algebraic expressions.
A crucial aspect of coordinate geometry is plotting points on a graph using their \((x, y)\) coordinates, helping visualize mathematical concepts like distance, relationships, and slopes.

• The x-axis runs horizontally, while the y-axis runs vertically.
• Quadrants divide the plane into four sections, typically marked I to IV, each with different characteristic sign combinations of \(x\) and \(y\) coordinates.

Understanding coordinate geometry is essential for determining the slope of a line.With our given points, we observe how the shift from \((-6,0)\) to \((0,6)\) translates into a visual slope on the graph indicated by our linear equation calculations.

Mathematics Education

Learning mathematics isn't just about solving problems — it's about grasping concepts and applying them in real-world contexts. Achieving this deeper understanding is a central goal of mathematics education.
To accomplish this:

• Break down complex problems into smaller, manageable steps, allowing students to tackle each part with confidence.
• Use visual aids like graphs to show how abstract concepts take shape in a real setting.

The slope of a line, for instance, serves as a critical teaching tool. Explaining its practical applications — such as comparing speeds or determining cost efficiency — helps students see its value beyond textbooks. This approach not only boosts comprehension but also encourages students to see the interconnectedness of mathematical ideas in daily life.