Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((5,-6)\) \(m=1\)

Step-by-Step Solution

Answer

The three additional points on the line are (6,-5), (7,-4) and (8,-3).

Step 1: Understand the slope

The slope, m, is defined as the rise over the run. In this instance, m is 1, which means that for each unit move to the right (increase x by 1), we also raise by one unit (increase y by 1).

Step 2: Find the first point

The given point is (5, -6). To find the next point, increase the x-coordinate by 1 unit, so x is now 6. Since the slope is 1, we also increase the y-coordinate, so y is now -5. The first new point is (6, -5).

Step 3: Find the second point

Use the first new point (6,-5). Increase the x-coordinate by 1 unit to get x = 7. With the slope of 1, we increase the y-coordinate by 1 unit to get y = -4. Hence, the second new point is (7, -4).

Step 4: Find the third point

Starting with the second new point (7,-4), increase the x-coordinate by 1 unit to get x = 8. With the slope of 1, also increase the y-coordinate by 1 unit to get y = -3. Hence, the third new point is (8,-3).

Key Concepts

Linear Equations

Linear equations describe a straight line on a coordinate plane. They express the relationship between the x and y coordinates. Usually, they are in the form \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, or where the line crosses the y-axis.

The equation helps us understand how the y-value changes as the x-value changes. In the problem you're working on, the slope \(m\) is 1. This means that for every increase in x, y increases by exactly the same amount. This consistent change creates a straight, diagonal line across the graph.

Coordinate Geometry

Coordinate geometry is a way to use algebra to study geometry. It involves plotting points, lines, and shapes on a coordinate plane. The plane consists of two axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

A point on the plane is described using two numbers (x, y), known as coordinates. In this exercise, the point (5, -6) is a starting point on the line. With these coordinates, you can plot the point on the graph. By applying the slope, you can determine how the line continues through the plane.

Finding Points on a Line

To find further points on a line, use the starting point and the slope. Start with the given point and adjust based on the slope to find new points. Here's how:

• Start at your known point, like (5, -6).
• Increase or decrease the x-coordinate based on your desired direction (right or left).
• Adjust the y-coordinate using the slope. For a slope of 1, both x and y increase equally.

In this problem, by repeatedly increasing x by 1 and y by 1, you discover the points (6, -5), (7, -4), and (8, -3). This method helps visualize how the line extends both forward and backward.