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Chapter 1: Problem 39

Solve. \(b d=c+b a,\) for \(b\)

Short Answer

Expert verified
b = \frac{c}{d - a}

Step by step solution

01

Isolate the term with b on one side of the equation

To solve for b, first make sure all terms involving b are on one side of the equation. Start by subtracting b a from both sides of the equation: \[ b d - b a = c \]
02

Factor out b from the left side

Next, factor out b from the left side of the equation: \[ b (d - a) = c \]
03

Solve for b

Finally, divide both sides of the equation by (d - a) to isolate b: \[ b = \frac{c}{d - a} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Manipulation
In solving algebraic problems, our first step is often to manipulate the given equation to a simpler form. This involves moving terms around and performing operations that transform the equation without changing its equality.
In our example, we start with the equation:
\(bd = c + ba\).
To make it easier to work with, we need to get all the terms involving the variable we want to solve for on one side. This step is crucial as it simplifies our next operations and sets a clear path to the solution.
Think of it like organizing your desk before starting a project. Moving things around makes sure everything necessary is in place.
Isolating Variables
The next key concept is isolating the variable you're solving for. In our example, we want to solve for \(b\).
We first move \(ba\) to the left side by subtracting it from both sides:
\[bd - ba = c\].
Isolating means making our desired variable stand alone on one side, so the next steps involve getting \(b\) by itself.
It's like clearing everything off your desk except the book you need to read. This step sets the stage to directly solve for \(b\).
Factoring
Factoring refers to rewriting an expression as a product of its factors. In our problem, we see both terms on the left side involving \(b\). Factoring helps us to simplify these terms into a single product:
\[b(d - a) = c\].
Here, we factor out \(b\) from \(bd - ba\), making the left side a product of \(b\) and \(d - a\).
Factoring is very useful as it condenses expressions and makes them easier to handle. Imagine it like bundling scattered papers into one neat file.
Solving for a Variable
Our final concept is solving for the variable itself. Once the variable is isolated and factored appropriately, we can solve for it directly. Continuing from our factored equation:
\[b(d - a) = c\],
we solve for \(b\) by dividing both sides by \(d - a\):
\[b = \frac{c}{d - a} \].
This step involves straightforward arithmetic to get the variable alone. Think of it as the final step of your project, where every detail comes together to complete the task.

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