Problem 10 - Brief Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. \((-1,4)\) and \((5,1)\)

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{1}{2}\).

Step 1: Understand the Slope Formula

We determine the slope of a line given two points using 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 from Points

Assign \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (5, 1)\).

Step 3: Substitute Values into the Formula

Substitute the values into the slope formula: \( m = \frac{1 - 4}{5 - (-1)} \).

Step 4: Calculate the Differences in the Numerator and Denominator

Compute the differences: \(1 - 4 = -3\) and \(5 - (-1) = 5 + 1 = 6\).

Step 5: Simplify the Fraction

The slope \( m = \frac{-3}{6} \) can be simplified to \( m = \frac{-1}{2} \).

Key Concepts

Linear Equations

Linear equations are foundational in mathematics, describing straight lines on a graph. Each linear equation can be written in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis. To find the equation of a straight line, you need two crucial pieces of information: a point on the line, and the slope. The slope determines both the direction and the steepness of the line.

Understanding how to calculate the slope from two points is essential because it tells you whether a line is rising, falling, or remaining constant:

• If the slope \( m > 0 \), the line rises from left to right.
• If \( m < 0 \), the line falls from left to right.
• If \( m = 0 \), the line is horizontal, indicating a constant function.

Mastering linear equations is crucial for solving many mathematical problems, especially those involving predictions and trend analysis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic methods to solve geometric problems. This branch of mathematics allows you to visualize points, lines, and shapes on a plane using a coordinate system. The system is defined by an "x-axis" and a "y-axis," intersecting perpendicularly at the origin (0,0).

In the context of our problem, we plotted two points:

• Point A: \((-1,4)\)
• Point B: \((5,1)\)

By using these coordinates, we calculated the slope of the line. These numerical values help us determine the position and the equation of the line in space. By finding the slope, you bridge the gap between abstract algebraic concepts and tangible geometric representations, making coordinate geometry a powerful tool for visualizing mathematical ideas.

Algebra

Algebra is a broad field of mathematics dealing with symbols and the rules for manipulating those symbols. It helps in structuring expressions, forming equations, and solving them. In the calculation of the slope in our exercise, we use basic algebraic manipulation techniques.

Through algebra, expressions like \( m = \frac{y_2 - y_1}{x_2 - x_1} \) are formed and simplified to find precise values. Here, the subtraction of coordinates \((y_2 - y_1)\) and \((x_2 - x_1)\) helps us find how much the line moves up or down for a certain shift left or right. In the given exercise:

• We subtracted 4 from 1 to get -3.
• Subtracted -1 from 5 (which means adding 1) to get 6.

Finally, simplifying the fraction from \(-\frac{3}{6}\) to \(-\frac{1}{2}\) illustrates algebra's power to reveal simpler, equivalent equations.

Such algebraic handling lays the groundwork for more complex mathematical tasks and problem-solving techniques.