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Chapter 2: Problem 68

Solve each formula for the given letter. $$ a c=b c+d, \text { for } c $$

Short Answer

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

Step by step solution

01

- Identify the given equation

The given equation is \(ac = bc + d\). We need to solve for \(c\).
02

- Isolate terms involving c

Move all terms involving \(c\) to one side of the equation. Subtract \(bc\) from both sides: \[ac - bc = d\].
03

- Factor out c

Factor \(c\) out of the left side of the equation: \[c(a - b) = d\].
04

- Solve for c

Divide both sides by \(a - b\) to solve for \(c\): \[c = \frac{d}{a - b}\].

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Key Concepts

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

Isolating Variables
When solving an equation, the first step is often to isolate the variable you are solving for on one side of the equation. In our exercise, we are solving for the variable \(c\) in the equation \(ac = bc + d\).
To isolate \(c\), start by getting all the terms that include \(c\) on one side of the equation. Here, we subtract \(bc\) from both sides:
\[ac - bc = d\]
This manipulation is crucial because it allows us to isolate the variable of interest, setting up the equation for the next step, which involves factoring.
Factoring
Factoring is a process used to simplify expressions or solve equations. It involves writing a number or an expression as a product of its factors. In our case, we have the equation \[ac - bc = d\], and we want to factor out the common variable, which is \(c\). So, we rewrite the left side:

\[c(a - b) = d\]
By factoring, we simplify the equation and make it easier to solve for the variable \(c\). Notice how \(c\) is now a single entity multiplied by another term \( (a - b)\). Factoring is a valuable tool in algebra that can make solving equations more straightforward.
Algebraic Manipulation
Algebraic manipulation encompasses a variety of techniques used to restructure an equation or expression. In our exercise, after isolating and factoring, we use algebraic manipulation to solve for \(c\). With the factored equation
\[c(a - b) = d\],
the next step is to isolate \(c\). We do this by dividing both sides of the equation by \( (a - b)\):
\[ c = \frac{d}{a - b}\]
This step completes our solution. Dividing by \( (a - b)\) isolates \(c\), giving us the final answer. Algebraic manipulation is all about rearranging the equation to solve for the desired variable, and involves skills like adding, subtracting, multiplying, dividing, and factoring.

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