Problem 71 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe how to write the equation of a line if its slope and a point on the line are known.

Step-by-Step Solution

Answer

The steps to writing the equation of a line with a known slope and a point are: 1. Understand the point-slope form of the equation of a line 2. Substitute the known slope and point into this formula. The resulting equation is the equation of the line.

Step 1: Understand the Point-Slope form

The point-slope form of a line is: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line and \( (x_1, y_1)\) are the coordinates of a point on the line.

Step 2: Substitute the known values

Suppose the known slope is \(m'\) and the known point is \((x'_1, y'_1)\). Substitute these into the point-slope form to get your equation of the line: \( y - y'_1 = m'(x - x'_1)\)

Key Concepts

Point-Slope Form

The point-slope form is a crucial concept in coordinate geometry for describing linear equations. It is particularly useful when you have a slope and a single point on a line. The standard equation is given by:

• \(y - y_1 = m(x - x_1)\)

In this formula:

• \(m\) represents the slope of the line
• \((x_1, y_1)\) are the coordinates of a known point on the line

The beauty of the point-slope form lies in its simplicity and flexibility. You can quickly form the equation of a line by plugging in the slope and any point along that line.
For example, if you know the slope \(m'\) and a specific point \((x'_1, y'_1)\), you can substitute these values into the formula as follows:

• \(y - y'_1 = m'(x - x'_1)\)

This represents the line in terms of the given slope and point.

Slope

The 'slope' of a line is an important feature in mathematics that helps you understand the line's direction and steepness.

• If the slope is positive, the line rises as it moves from left to right.
• If it's negative, the line falls as it moves from left to right.
• A slope of zero means the line is completely horizontal.
• An undefined slope indicates a vertical line.

The slope \(m\) is typically calculated using two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\). The formula for calculating the slope is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

With this method, you can determine the exact tilt of any line in the coordinate plane. Understanding the slope allows you to visualize how a line behaves without needing to see it graphically.
This foundational concept is invaluable when dealing with linear equations and is a key player in the point-slope form.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is where algebra meets geometry through graphs and coordinates. It allows you to describe geometric shapes, like lines, and their properties using the coordinate plane. This approach provides a powerful means to solve geometric problems with algebraic methods.

• Coordinates are typically described by an \((x, y)\) pair on the Cartesian plane.
• Graphs and lines are drawn based on these coordinate pairs.

This methodology enables the representation of equations, like the point-slope form, graphically.
For instance, when you use the point-slope form \(y - y_1 = m(x - x_1)\), you plot it in a grid where each point corresponds to coordinates on the plane.

• The intersection of the x-axis and y-axis acts as the origin, or point \((0, 0)\).
• Lines can thus be analyzed, allowing calculations of distances, midpoints, and, notably, slopes.

By visualizing these elements, coordinate geometry makes complex problems more tangible and solutions more intuitive.