Problem 28 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ t=t_{0}+\frac{k}{2} w, \text { for } w $$

Step-by-Step Solution

Answer

Question: Solve the equation $$t = t_{0} + \frac{k}{2} w$$ for the variable $$w$$. Answer: $$w = \frac{2(t - t_{0})}{k}$$

Step 1: Identify the term containing the variable w

In the given equation, the term containing the variable $$w$$ is $$\frac{k}{2} w$$.

Step 2: Subtract $$t_{0}$$ from both sides of the equation

To isolate the term containing $$w$$, subtract $$t_{0}$$ from both sides of the equation as shown below: $$t - t_{0} = \frac{k}{2} w$$

Step 3: Divide by the constant coefficient

The term containing $$w$$ has a constant coefficient of $$\frac{k}{2}$$. To solve for $$w$$, divide both sides of the equation by this coefficient: $$\frac{t - t_{0}}{\frac{k}{2}} = w$$

Step 4: Simplify the expression

To simplify the expression, multiply the numerator and denominator by 2: $$w = \frac{2(t - t_{0})}{k}$$ Now, the equation is solved for the variable $$w$$, and the final solution is: $$w = \frac{2(t - t_{0})}{k}$$

Key Concepts

Variable Isolation

One of the cornerstones of solving algebraic equations is the concept of variable isolation. This means getting the variable you are solving for on one side of the equation by itself. In more intuitive terms, when you're asked to "solve for \( w \)" in an equation, it means you want \( w \) on one side, standing alone without any other terms attached to it.
In our given problem, we start with:

• \( t = t_{0} + \frac{k}{2} w \)

We see \( w \) is not isolated due to the presence of \( t_{0} \) and the coefficient \( \frac{k}{2} \). To isolate \( w \), we need to "move" everything else to the other side of the equation. This typically involves operations like addition, subtraction, multiplication, or division.
In this problem, our first step is to subtract \( t_{0} \) from both sides. This narrows down the focus towards \( w \), leaving us with:

• \( t - t_{0} = \frac{k}{2} w \)

Variable isolation often sets the stage for the next part of the solution, which is equation solving.

Equation Solving

Once the variable is isolated as much as possible or practical, the next step is to solve the equation. Solving means performing operations that leave the variable, in this case, \( w \), by itself.
In our present equation:

• \( t - t_{0} = \frac{k}{2} w \)

We still have a coefficient \( \frac{k}{2} \) attached to \( w \). To "get rid of" this coefficient and completely isolate \( w \), we need to do the opposite operation of what is currently being done to it.
Since \( w \) is being multiplied by \( \frac{k}{2} \), we divide both sides by this coefficient:

• \( \frac{t - t_{0}}{\frac{k}{2}} = w \)

This division detaches the coefficient, bringing us another step closer to \( w \) being fully isolated and the equation solved.

Expression Simplification

Simplification often follows after the major logical steps and involves making the solution as clear and concise as possible. Here, we want to simplify the fraction obtained in the previous step.

• \( \frac{t - t_{0}}{\frac{k}{2}} = w \)

Handling a fraction divided by another fraction can sometimes be trickier. To simplify, it's useful to multiply both the numerator and divisor by the same value to get rid of fractions within fractions.
Here, multiplying both by 2 results in a clearer expression:

• \( w = \frac{2(t - t_{0})}{k} \)

Now the equation is simplified neatly, showing \( w \) as an isolated variable on one side. Simplification is all about making expressions more manageable and user-friendly, which makes the final answer easier to understand and use in further calculations.