Problem 30 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ a b+a w=c-a w, \text { for } w . $$

Step-by-Step Solution

Answer

Question: Given the equation ab + aw = c - aw, solve for w in terms of a, b, and c. Answer: w = (c - ab) / (2a)

Step 1: Setup the equation

We begin by writing down the given equation: $$ ab + aw = c - aw $$

Step 2: Move the terms containing w to one side

To isolate w, move the terms with w to one side of the equation. Add aw to both sides of the equation to achieve this. $$ ab + aw + aw = c - aw + aw $$

Step 3: Combine the w terms

Now we simplify the equation by combining the w terms on the left side of the equation. $$ ab + 2aw = c $$

Step 4: Isolate w

To solve for w, we need to isolate it. Divide both sides of the equation by the factor of w, which is 2a. $$ w = \frac{c - ab}{2a} $$ Now we have found the solution for w in terms of a, b, and c: $$ w = \frac{c - ab}{2a} $$

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra. It involves finding the value of an unknown variable that satisfies the given equation. In our example, we start with the equation: \[ ab + aw = c - aw \]The goal is to manipulate the equation until we can clearly see the solution. This requires performing operations that simplify the equation step by step, ensuring each operation is valid and maintains balance on both sides of the equation. By doing so, we can gradually uncover the value of the unknown variable. In this case, we're focusing on solving for \( w \), ultimately expressing it in terms of the other constants \( a \), \( b \), and \( c \). Some typical operations in solving equations include:

• Adding or subtracting the same quantity on both sides.
• Multiplying or dividing by a non-zero constant.
• Rearranging terms to simplify the equation further.

These operations preserve the equality, helping us move closer to the solution.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable is alone on one side. This process allows us to see what the variable equals in terms of the other quantities present in the equation. In our example, we need to solve for \( w \), so we focus on manipulating the equation:\[ ab + aw = c - aw \]Initially, \( w \) is present on both sides. To isolate it, we decide to move all terms involving \( w \) to one side. We do this by adding \( aw \) to each side:\[ ab + aw + aw = c - aw + aw \]This results in:\[ ab + 2aw = c \]Now, \( w \) is only on the left side, and we can further isolate it by dividing both sides by \( 2a \) (since this is the coefficient in front of \( w \)). This operation will provide the value of \( w \) directly:\[ w = \frac{c - ab}{2a} \]This effectively isolates \( w \), showing its relationship with the constants \( a \), \( b \), and \( c \).

Combining Like Terms

Combining like terms is a strategy used in algebra to simplify equations by merging terms that have the same variables. In the process of solving equations, this step helps reduce the complexity of the expression.In the given equation:\[ ab + aw + aw = c \]You'll notice that the terms \( aw \) and \( aw \) are like terms because they both contain the variable \( w \). Therefore, they can be combined to form\[ ab + 2aw = c \]This combination simplifies the equation by condensing it into fewer terms. It makes the next steps in solving for \( w \) straightforward. By ensuring that the 'like' variables or terms are combined, we create an expression that's easier to manipulate and solve. This simplification is crucial for managing more complex algebraic equations efficiently. It's a building block for algebraic manipulation, helping students refine their skills in equation solving.