Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the line that passes through the two given points. (1,5) and (4,11)

Step-by-Step Solution

Answer

The slope is 2.

Step 1: Identify the Points

We have two points given: Point A as \((x_1, y_1) = (1, 5)\) and Point B as \((x_2, y_2) = (4, 11)\). The task is to find the slope of the line that passes through these points.

Step 2: Apply the Slope Formula

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the values for the points into the formula.

Step 3: Perform the Calculation

Substituting the values from Points A and B into the slope formula: \[ m = \frac{11 - 5}{4 - 1} \]Simplifying the numerator and the denominator gives:\[ m = \frac{6}{3} \]

Step 4: Simplify the Result

Divide 6 by 3 to simplify the slope:\[ m = 2 \]Hence, the slope of the line is 2.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study geometric figures using a coordinate system. It helps us to analyze and understand the spatial relationships and properties of shapes and sizes in a plane. Using coordinate geometry, we can answer questions about distances, angles, and slopes, which are key elements when working with points and lines.

Coordinate geometry operates within a plane, usually represented by a two-dimensional surface called the Cartesian plane. The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which has coordinates (0, 0).

In this coordinate system, any point can be represented by a pair of numerical coordinates. These coordinates describe the point's location relative to the two axes. For instance, in the exercise given, (1, 5) and (4, 11) are the coordinates of two points in the Cartesian plane. The first number in each pair is the x-coordinate, and the second is the y-coordinate. By understanding these coordinates, you can easily define the position of points in a plane.

Slope Formula

The slope formula is an essential tool in coordinate geometry for determining how steep a line is. The slope of a line is a measure of its incline or decline and is represented by the letter \( m \). In practical terms, the slope tells us how much the line rises or falls vertically for each unit of horizontal movement.

The formula for calculating the slope \( m \) when you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), on a plane is:

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

This formula uses the differences in the y-coordinates and x-coordinates of the two points.

• The numerator \((y_2 - y_1)\) represents the change in the vertical direction (rise).
• The denominator \((x_2 - x_1)\) represents the change in the horizontal direction (run).

To find the slope, simply substitute the coordinates into this formula and perform the calculations to obtain a numerical value, as we did in our example using points (1, 5) and (4, 11).

Points on a Plane

Points on a plane are the basic elements of coordinate geometry. Understanding how to work with them is crucial. Each point on a plane is defined by a coordinate pair \((x, y)\), which tells us exactly where it is located in relation to the x-axis and the y-axis of the Cartesian coordinate system.

For example, the point \((1, 5)\) indicates:

• The point is located at 1 unit along the x-axis, starting from the origin.
• It is also 5 units along the y-axis.

Similarly, the point \((4, 11)\) provides another precise location on the plane. When trying to find the slope between two points — such as in the given exercise — it is essential to correctly identify and use their coordinates. This helps in accurately calculating slope and understanding how one point relates to another in terms of position.

By manipulating these points, you can draw lines, determine distances, and calculate slopes, enriching your understanding of geometric relationships.