Problem 10 - 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? $$y=m x+b \text { for } x$$

Step-by-Step Solution

Answer

(y-b)/m

Step 1: Isolate 'x' term

First, we move the term not containing 'x' to the other side of the equation. In this case, subtract 'b' from both sides to get the equation as \(y - b = m x\)

Step 2: Solve for 'x'

Next, we divide the entire equation by 'm' to solve for 'x'. So that gives us \(x = (y-b)/m\)

Key Concepts

Linear Equations

Linear equations are among the most fundamental types of equations in algebra. These equations form the backbone for many real-world applications, including economics, engineering, and technology. A linear equation is essentially an expression where the highest power of the variable is one. This creates a straight line when graphed on a coordinate plane.

Standard form of a linear equation is usually written as \(y = mx + b\) where:

• \(y\) and \(x\) are variables.
• \(m\) is the slope of the line, showing the steepness and direction.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

To solve linear equations, the goal is often to find the value of one variable with respect to others. Recognizing these forms and being comfortable manipulating them is key to mastering algebraic problems.

Variable Isolation

Variable isolation is a crucial technique in solving equations. It involves rearranging equations so that a particular variable stands alone on one side. This process turns the complex problem into a simpler one, where the variable is expressed in terms of known values or other variables. Consider our goal in the original exercise: solving for \(x\) in the equation \(y = mx + b\).

To isolate \(x\), we carry out the following steps:

• First, subtract \(b\) from both sides to move anything not involving \(x\) away from it. This step transforms the equation into \(y - b = mx\).
• Next, divide the entire equation by \(m\) to completely isolate \(x\). Now we have \(x = \frac{y-b}{m}\).

Remember that the aim of variable isolation is to simplify problem-solving by focusing on a single variable. It is a widely used technique in various fields to find unknown quantities.

Formula Manipulation

Formula manipulation is the art of altering equations to simplify them or reveal more information. The ability to strategically rewrite formulas is an essential skill in mathematics. This skill includes adding, subtracting, multiplying, or dividing both sides of an equation to maintain its integrity and balance.

The original problem is a classic example of formula manipulation. By:

• Subtracting \(b\) from both sides, we shift parts of the equation to simplify \(y = mx + b\) into \(y - b = mx\).
• Dividing by \(m\), we further adjust the equation to solve for \(x\), yielding \(x = \frac{y-b}{m}\).

These steps not only isolate the desired variable but also maintain the equation's balance. Understanding how to manipulate equations opens many pathways to solving complex mathematical problems and is foundational in calculus, physics, and higher-level computing. Practice this regularly to gain confidence and efficiency in solving algebraic challenges.