Problem 70 - Elementary Algebra

Question

For the following problems, solve each literal equation for the designated letter. \(K=\frac{1}{2} h\left(s_{1}+s_{2}\right)\) for \(s_{2}\)

Step-by-Step Solution

Answer

Question: Given the equation \(K=\frac{1}{2} h\left(s_{1}+s_{2}\right)\), find the expression for \(s_{2}\). Answer: \(s_{2}=\frac{2K}{h}-s_{1}\)

Step 1: Isolate the term with \(s_{2}\)

Divide both sides of the equation by \(\frac{1}{2}h\): $$\frac{K}{\frac{1}{2}h}=\frac{\frac{1}{2}h\left(s_{1}+s_{2}\right)}{\frac{1}{2}h}$$ This simplifies to: $$\frac{2K}{h}=s_{1}+s_{2}$$

Step 2: Solve for \(s_{2}\)

Now subtract \(s_{1}\) from both sides of the equation to isolate \(s_{2}\): $$\frac{2K}{h}-s_{1}=s_{2}$$ So, the expression for \(s_{2}\) is: $$s_{2}=\frac{2K}{h}-s_{1}$$

Key Concepts

Solving Equations

When we talk about solving equations, we refer to the process of finding the value of an unknown variable that makes the equation true. In this particular exercise, we are dealing with a **literal equation** which means it contains several variables. Our task is to express one of these variables, specifically \(s_2\), in terms of the other variables given.
The process often involves systematically rearranging the equation. This allows us to isolate the desired variable. For example, in the equation \(K=\frac{1}{2} h(s_{1}+s_{2})\), our goal is to rearrange it so that \(s_2\) is on one side of the equation by itself.
Steps often include techniques such as:

• Division or multiplication to eliminate coefficients.
• Adding or subtracting terms on both sides of the equation.

These operations help us manipulate the equation to solve for the unknown.

Isolating Variables

Isolating a variable is a fundamental skill in algebra that involves moving all instances of a variable to one side of the equation. Consider the objective of solving for \(s_2\). Initially, this term is blended with \(s_1\) within the equation \(K=\frac{1}{2} h(s_{1}+s_{2})\).
To isolate \(s_2\), the first step is to eliminate the fraction by dividing both sides by \(\frac{1}{2}h\). This action gives the simpler form \(\frac{2K}{h}=s_{1}+s_{2}\). Notice how \(s_2\) is now on one side but not yet isolated.
Next, another isolation technique is applied: subtract \(s_1\) from both sides, resulting in \(s_{2}=\frac{2K}{h}-s_{1}\). Keep in mind that when isolating variables, maintaining balance by performing the same operation on both sides is crucial.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations using algebra basics to simplify or solve them for a specific variable. In our work on the equation \(K=\frac{1}{2} h(s_{1}+s_{2})\), we performed a series of manipulations.
First, we needed to clear out the fraction, which we achieved by multiplying by the reciprocal, \(\frac{2}{h}\). This operation allowed us to simplify our equation. For example, moving from \(K\) to \(\frac{2K}{h}\) involves multiplication, a key algebraic operation.
Subtraction is then used to separate terms. By subtracting \(s_1\) from both sides, we effectively isolated \(s_2\). Algebraic manipulations like these, involving basic operations such as addition, subtraction, multiplication, and division, form the core of working through literal equations.