Problem 8 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (3,-1) \text { and }(5,7) $$

Step-by-Step Solution

Answer

The slope of the line is 4.

Step 1: Identify the Points

We are given two points on a Cartesian plane, \((x_1, y_1) = (3, -1)\) and \((x_2, y_2) = (5, 7)\). In order to find the slope of the line passing through these points, we need to use these coordinates in the slope formula.

Step 2: Recall the Slope Formula

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula calculates the "rise over run" or vertical change divided by horizontal change.

Step 3: Substitute Values into the Formula

Plug the coordinates of the points \((3, -1)\) and \((5, 7)\) into the slope formula: \[ m = \frac{7 - (-1)}{5 - 3} \].Perform the arithmetic operations in the numerator and denominator.

Step 4: Calculate the Slope

Calculate the difference in the \(y\)-coordinates: \(7 - (-1) = 7 + 1 = 8\). Calculate the difference in the \(x\)-coordinates: \(5 - 3 = 2\). Now substitute these values back into the slope formula: \[ m = \frac{8}{2} = 4 \].

Step 5: State the Slope

The slope of the line determined by the points \((3, -1)\) and \((5, 7)\) is \(4\). Therefore, the slope is defined and it indicates a line that rises \(4\) units for each unit it moves to the right.

Key Concepts

Point-slope formula

The point-slope formula is an essential concept when finding the equation of a line or understanding its slope. This formula is particularly useful if you already know a point on the line and the slope of the line.

The formula is written as: \( y - y_1 = m(x - x_1) \) Where:

• \(m\) is the slope of the line.
• \((x_1, y_1)\) is the point on the line.

Using this formula helps you derive the equation of the line. This process involves substituting the point coordinates and the slope \(m\) into the formula. It's important to understand that this is just one way to express the equation of a line, but it's extremely helpful when you have specific points recorded.

Coordinate geometry

Coordinate geometry is a branch of mathematics that helps us understand spatial relationships using numbers. It relies heavily on the use of coordinates to analyze geometric shapes and figures.

• It allows you to calculate distances, angles, and other properties of geometric shapes.
• You can plot points on a Cartesian plane, which is a system of defining precise locations in a 2D space.
• Different equations can be used to describe lines, circles, and more using coordinate geometry.

By using precise numerical coordinates, you gain a deeper level of analytical capability and visualization of geometric properties. These applications are found widely in physics, engineering, and computer graphics.

Cartesian plane

The Cartesian plane is a two-dimensional surface where we can plot points, lines, and curves using ordered pairs (coordinates). Named after René Descartes, this system revolutionized geometry by integrating algebra and geometry together.A Cartesian plane is defined by two perpendicular number lines:

• The horizontal line, known as the x-axis.
• The vertical line, known as the y-axis.

Where these axes intersect is called the origin, denoted as (0,0). Each point on the Cartesian plane can be described using coordinates \((x, y)\). Here, \(x\) is the horizontal position, and \(y\) is the vertical position. Understanding this allows you to navigate and represent points and shapes accurately in a 2D space.

Arithmetic operations

Arithmetic operations are basic mathematical operations essential for problem-solving in many areas, including coordinate geometry. These operations consist of addition, subtraction, multiplication, and division. To find the slope of a line or perform other calculations in geometry:

• **Addition/Subtraction:** Use these to compute the rise and run by determining the difference between coordinates.
• **Division:** Employ division to find the slope by dividing the rise by the run.

Through these operations, you can manipulate numbers and variables, facilitating a wide range of calculations. Whether you're adjusting equations or solving for variables, arithmetic operations form the foundation of mathematical logic and reasoning.