Problem 40 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ E=m c^{2} \text { for } m $$

Step-by-Step Solution

Answer

The solution for \(m\) is \(m = \frac{E}{c^2}\).

Step 1: Identify the Goal

The goal is to solve the equation \(E = m c^2\) for the variable \(m\). This means we need to rearrange the equation to express \(m\) in terms of \(E\) and \(c\).

Step 2: Isolate the Variable

We start with the equation \(E = m c^2\). To isolate \(m\), divide both sides of the equation by \(c^2\) to get:\[m = \frac{E}{c^2}\]

Step 3: Verification

To verify our solution, substitute \(m = \frac{E}{c^2}\) back into the original equation to check whether it holds true as \(E = m c^2\). By plugging in, we see that \(m c^2 = \frac{E}{c^2} \cdot c^2 = E\), thus verifying our solution.

Key Concepts

Understanding Algebra

Algebra is an essential field of mathematics that uses symbols, usually letters, to represent numbers or quantities in equations and formulas. The power of algebra lies in its ability to abstractly represent problems and find unknown values using logical reasoning. For example, in the equation \(E = mc^2\), each symbol holds a specific meaning: \(E\) stands for energy, \(m\) for mass, and \(c\) for the speed of light.

• Variables: Symbols like \(m\), \(E\), and \(c\) are called variables. They can represent known or unknown values.
• Equations: An equation states that two expressions are equal. It usually includes an equals sign (\(=\)).

In algebra, the main goal often involves finding the unknown variable, such as finding \(m\) when given \(E\) and \(c\). This requires manipulating the equation systematically.

Isolation of Variables

Isolation of variables is a fundamental technique in solving algebraic equations. It involves rearranging the equation to have the unknown variable by itself on one side. In this process, we apply inverse operations to both sides of the equation to maintain balance.For instance, consider the equation \(E = mc^2\). To isolate \(m\), we identify that \(m\) is multiplied by \(c^2\). To isolate \(m\), we need to divide both sides by \(c^2\):\[ m = \frac{E}{c^2} \]

• Identify the operations involved with the variable.
• Apply the inverse operation to both sides.

By isolating \(m\), we can express \(m\) explicitly in terms of the other variables, making it the subject of the formula.

Formula Manipulation

Formula manipulation is about changing the appearance of an equation without altering its underlying relationships. This skill is vital when working with equations in science, engineering, and math. Manipulating formulas often involves steps like addition, subtraction, multiplication, or division, and understanding these operations is key.In our example, we manipulated the original formula \(E = mc^2\) to solve for \(m\). Here’s how we did it:

• We recognized that \(m\) was multiplied by \(c^2\).
• We divided both sides of the equation by \(c^2\) to "move" it from one side to the other.

After this manipulation, our formula becomes \(m = \frac{E}{c^2}\), which is mathematically equivalent to our original formula but restructured to show \(m\) as the dependent variable. Understanding these transformations provides a foundation for tackling more complex problems across different disciplines.