Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$y=m x+b \text { for } m$$

Step-by-Step Solution

Answer

The formula solved for 'm' is \(m = (y - b) / x\).

Step 1: Understanding the formula

The equation given is in the form \(y = mx + b\), which is the standard form for a linear equation where 'm' is the slope of the line and 'b' is the y-intercept.

Step 2: Rearranging to solve for 'm'

To isolate 'm', we need to rearrange the equation by subtracting 'b' from both sides which gives us: \(y - b = mx\).

Step 3: Solving for 'm'

Finally, to find 'm', we need to divide both sides by 'x'. This gives us the equation in terms of 'm': \(m = (y - b) / x\).

Key Concepts

Linear Equations

A linear equation is a type of algebraic equation in which each term is either a constant or the product of a constant and a single variable. The most basic form you will usually come across is the slope-intercept form: \(y = mx + b\). This form represents a straight line when plotted on a graph, and is characterized by its simplicity and predictability.

Linear equations are foundational in algebra as they model a constant rate of change. In the context of the exercise, the formula \(y = mx + b\) expresses the relationship between \(x\) and \(y\), matching the format of a line on a graph.

Learning to recognize and manipulate linear equations is crucial because they appear everywhere in math and science. Whether you're dealing with a phone plan, calculating grades, or figuring out a budget, linear equations help forecast and describe relationships.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or expressions to isolate variables or to express them in a different form. In our example, we're tasked with solving the equation \(y = mx + b\) for \(m\).

To achieve this, you follow a series of logical and reversible steps. Here’s a quick guide to manipulate such equations:

• First, identify the variable you need to isolate—in this case, \(m\).
• Next, isolate terms involving \(m\) by performing inverse operations. Here, we subtract \(b\) from both sides, resulting in \(y - b = mx\).
• Finally, divide each side by \(x\) to solve for \(m\), leading to the solution \(m = \frac{y-b}{x}\).

Each of these steps involves careful balancing of both sides of the equation, a fundamental principle in algebra. Mastering algebraic manipulation allows better problem-solving skills across different math areas.

Slope and Intercept

The slope-intercept form of a linear equation \(y = mx + b\) gives two essential pieces of information about a line: the slope \(m\) and the y-intercept \(b\). Understanding these elements is key to graphing and interpreting linear relationships.

- **Slope \(m\)**: Indicates the steepness and direction of a line. It tells us how much \(y\) changes for a change in \(x\). A positive slope means the line rises, while a negative slope indicates it falls.- **Intercept \(b\)**: Describes where the line crosses the y-axis. It represents the value of \(y\) when \(x\) is zero.

• Understanding these concepts aids in predicting and graphing lines.
• The slope helps in analyzing trends, while the intercept provides a starting point in the graph.

These components together make the linear equation invaluable for visualizing data and trends. They simplify complex relationships into clear, understandable visuals.