Problem 71 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(Q=\frac{2 m n}{s+t}\) for \(s\)

Step-by-Step Solution

Answer

Question: Solve the given literal equation for s: \(Q=\frac{2 m n}{s+t}\) Answer: \(s = \frac{2mn - Qt}{Q}\)

Step 1: Write down the given equation

First, we will write down the given equation: \(Q=\frac{2 m n}{s+t}\)

Step 2: Isolate s

Next, we'll isolate the s on one side of the equation. Start by multiplying both sides by (s + t) to eliminate the denominator: \(Q (s + t) = 2mn\)

Step 3: Distribute Q

Now, distribute Q on the left side of the equation: \(Qs + Qt = 2mn\)

Step 4: Isolate s further

Continue isolating s by subtracting Qt from both sides of the equation: \(Qs = 2mn - Qt\)

Step 5: Solve for s

Finally, divide both sides of the equation by Q to solve for s: \(s = \frac{2mn - Qt}{Q}\)

Step 6: Provide the final answer

After solving the equation, we have the final expression for s: \(s = \frac{2mn - Qt}{Q}\)

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving mathematical problems. It involves the use of various algebraic techniques to rearrange and simplify equations or expressions. The core of algebraic manipulation lies in understanding and applying properties such as the distributive property, associative property, and commutative property.

For example, in the provided exercise, distributing the variable Q across (s + t) and then isolating the variable of interest are forms of algebraic manipulation. These steps enable us to simplify complex expressions and solve for variables.

Here are some common algebraic manipulation techniques:

• Distributing a factor over a sum or difference (e.g. a(b + c) = ab + ac)
• Combining like terms
• Factoring out a common factor
• Using inverse operations to move terms from one side of an equation to the other

Mastering algebraic manipulation is essential for tackling a wide range of problems in algebra, and it serves as the foundation for more advanced mathematical concepts.

Isolating Variables

Isolating variables is a crucial step in solving equations, especially when dealing with literal equations where the goal is to express one variable in terms of others. The approach focuses on getting a single variable by itself on one side of an equation with all other terms on the opposite side.

The process often includes adding, subtracting, multiplying, or dividing both sides of the equation by the same value or expression. This ensures that the equation remains balanced. In the exercise example, isolating the variable s involves multiplying both sides by (s + t), distributing Q, and then subtracting Qt from both sides.

When isolating a variable, here are some tips to consider:

• Perform actions that simplify the equation, like reducing fractions or combining like terms.
• Use inverse operations that 'undo' each other (e.g., addition with subtraction).
• Keep the goal in mind: aim to have the variable you're solving for alone on one side.

Understanding how to isolate variables paves the way to finding solutions to equations efficiently.

Literal Equations

Literal equations are equations where variables represent quantities. They become particularly useful when there's a need to solve for one variable in terms of others. This skill applies to various fields such as physics, engineering, and economics, where formulas are reshaped depending on what quantity needs to be calculated.

In the context of our exercise, the equation given is a literal equation:
\(Q=\frac{2 m n}{s+t}\). Here, we are asked to solve for s in terms of Q, m, n, and t.

Steps in solving literal equations might include:

• Identify the variable to solve for.
• Use algebraic manipulation to rearrange the equation.
• Isolate the chosen variable by performing operations that reverse what's been applied to the variable.
• Check the solution by plugging it back into the original equation.

Understanding how to work with literal equations enhances problem-solving skills and offers insights into how different variables are interconnected.