Problem 14 - Algebra Form and Function

Question

Write an equation in point-slope form for the line. Through (8,10) with slope \(m=-3\)

Step-by-Step Solution

Answer

Question: Write the equation of the line in point-slope form that passes through the point (8, 10) and has a slope of -3. Answer: y - 10 = -3(x - 8)

Step 1: Identify the given point and slope

We are given a point (8, 10) and the slope \(m=-3\). We can identify \(x_1 = 8\), \(y_1 = 10\) and \(m = -3\).

Step 2: Plug the values into the point-slope form

Now, plug the values of \(x_1, y_1\), and \(m\) into the point-slope form equation: \(y - y_1 = m(x - x_1)\). It becomes: \(y - 10 = -3(x - 8)\).

Step 3: Simplify the equation

The equation in point-slope form is already simplified, so there is no further simplification needed. The final equation is: \(y - 10 = -3(x - 8)\).

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra and coordinate geometry. It allows us to describe a straight line using a mathematical expression. Essentially, an equation of a line expresses the relationship between the x-coordinates and y-coordinates of any point on the line.

One popular form is the point-slope form, utilized when a point on the line and the slope are given. The point-slope form is written as:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) represents the slope of the line, while \( (x_1, y_1) \) are the coordinates of a given point on the line.

This form is particularly useful as it immediately incorporates both the slope and a specific point. Once filled with the right values, it instantly provides the equation describing the entire line.

Slope

The slope of a line is a measure of its steepness, indicating how much the line ascends or descends as you move along it. In mathematics, slope is denoted by the letter \( m \).

To calculate the slope, you'll look at the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on the line. The formula for slope is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope indicates the line rises as it moves to the right, while a negative slope shows it falls. A zero slope illustrates a horizontal line, and an undefined slope signifies a vertical line.

Understanding slope is crucial, especially in interpreting how various lines behave in a coordinate plane. Slope forms the backbone of countless concepts in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate plane. This math field allows geometric problems to be represented algebraically and solved systematically.

In the coordinate plane, each point is defined by an ordered pair \((x, y)\), where \( x \) stands for the horizontal position and \( y \) the vertical. Lines, circles, and other geometric figures can thus be expressed using equations.

When working with lines, a key idea is the interpretation of the line's equation, such as identifying the slope and y-intercept. Coordinate geometry connects algebraic concepts to geometric visuals, providing a balanced approach to both disciplines. It equips users with tools to solve real-world problems and create graphical representations of mathematical relationships.