Problem 7 - 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? $$E=m c^{2} \text { for } m$$

Step-by-Step Solution

Answer

The mass \(m\) in the given formula can be rewritten as \(m=E/c^2\).

Step 1: Identify the target variable

The target variable that needs to be isolated is \(m\). In the equation \(E=mc^2\), \(m\) is currently multiplied by \(c^2\). So, the next step is to divide both sides of the equation by \(c^2\) to isolate \(m\).

Step 2: Perform the operation

By dividing both sides of the equation by \(c^2\), you get a new equation: \(m=E/c^2\). This new equation is now solved for \(m\).

Step 3: Verification

The equation \(m=E/c^2\) can be verified by substituting \(m\) from this equation into the original equation and checking if both sides of the equation are identical. This completes the process of solving the equation for \(m\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of one or more variables that make the equation true. In our original exercise, the equation given is Einstein's famous mass-energy equivalence formula: \(E = mc^2\). Here, "solving the equation" means rearranging it to express a specific variable, such as \(m\), in terms of the other quantities present.To start solving, you need to identify which variable you want to isolate, and then perform algebraic manipulations to express that variable in terms of the others.

• This typically involves operations such as addition, subtraction, multiplication, and division.
• Make sure to perform the same operation on both sides of the equation to maintain equality.

For instance, if \(m\) is multiplied by \(c^2\), you divide both sides by \(c^2\) to solve for \(m\). Successfully solving an equation allows you to calculate unknown quantities, which is essential for many applications in physics and mathematics.

Isolating Variables

Isolating variables is a technique used to rearrange equations to make one variable the subject. It's a vital concept whenever you want to express a variable in terms of other variables. For the equation \(E = mc^2\), isolating \(m\) involves making \(m\) appear on one side of the equation all by itself.To isolate \(m\), we perform inverse operations to undo what is currently being done to \(m\). In this case, since \(m\) is multiplied by \(c^2\), we'll do the opposite and divide both sides of the equation by \(c^2\):

• Divide both sides by \(c^2\): \(m = \frac{E}{c^2}\).
• Ensure every operation is applied equally to both sides to keep the equation balanced.

By doing this, we effectively "solve for" \(m\), making it isolated on one side of the equation, which is essential to determine its value based on known quantities of \(E\) and \(c\).

Physics Formulas

Physics formulas are mathematical representations of physical laws and principles. These formulas often involve multiple variables representing physical quantities like energy, mass, velocity, etc. The equation you explored, \(E = mc^2\), describes the relationship between energy \(E\), mass \(m\), and the speed of light \(c\).Such formulas are vital in physics because they allow predictions and computations related to real-world phenomena. Here’s what to remember:

• Formulas are derived based on empirical and theoretical principles from observed physical phenomena.
• Understanding how to manipulate these formulas by solving for different variables enables you to apply them flexibly to various scenarios.

In practical terms, the ability to rearrange and isolate variables in physics formulas, like \(E = mc^2\), equips you with tools to calculate quantities such as the amount of mass \(m\) required to produce a given amount of energy \(E\), given the constant speed of light \(c\). This powerful equation underlines the deep connection between mass and energy, a cornerstone of modern physics.