Problem 63 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(E=m c^{2}\) for \(m\)

Step-by-Step Solution

Answer

Question: Solve the equation \(E=mc^2\) for the variable \(m\). Answer: \(m=\frac{E}{c^2}\)

Step 1: Identify the given equation

The given equation is: \(E = mc^2\)

Step 2: Solve for 'm'

To isolate 'm', divide both sides of the equation by \(c^2\): \(m=\frac{E}{c^2}\)

Step 3: Final Solution

The literal equation, solved for \(m\), is: \(m=\frac{E}{c^2}\)

Key Concepts

Solving for a variable

When solving for a variable in a literal equation, your goal is to isolate that variable on one side of the equation. This process involves algebraically manipulating the equation, so that the variable of interest stands alone. Literal equations are equations where variables represent known or unknown values. They are often used in physics and other sciences to represent general relationships. In our example, the equation is Einstein's famous formula for energy, where you solve for the mass.

• First, identify the symbol for the variable you need to solve for, which, in this case, is \( m \).
• Next, perform operations to both sides of the equation to isolate that variable.

By isolating \( m \) using division, you've inverted the existing multiplication operation. Understanding how to rearrange equations is crucial for tackling more complex problems in mathematics and physics.

Algebraic manipulation

Algebraic manipulation is a key skill that allows us to solve equations for specific variables. This involves applying mathematical operations to both sides of the equation to make it simpler or to isolate one variable. A common technique is to use inverse operations. For example, if a variable is multiplied by something, you divide to cancel it out, and vice versa. This is exactly what we did with the equation \( E = mc^2 \):

• We wanted to get 'm' by itself, so we used division, as it's the inverse of multiplication with \( c^2 \).

When performing algebraic manipulations, each step must maintain the equality of the equation, meaning you must perform the same operation on both sides. Practicing these skills will make solving equations more intuitive over time.

Mathematical formula

The term 'mathematical formula' refers to a set equation that numerically describes a relationship between different variables. Formulas can look daunting, but they just convey simple mathematical relationships in a concise way. Take the famous formula \( E = mc^2 \), which shows how energy \( E \) is related to mass \( m \) and the speed of light \( c \):

• \( E \) is energy, often measured in joules.
• \( m \) is mass, typically in kilograms.
• \( c \) is the speed of light, a constant at approximately \( 3 \times 10^8 \) meters per second.

By understanding and using formulas, you can convert between different physical quantities or solve for unknowns, making them a critical tool in both everyday applications and advanced scientific research.