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Chapter 2: Problem 89

Solve each formula for the specified variable. $$ M=C(1+r) \text { for } r $$

Short Answer

Expert verified
r=\frac{M}{C}-1

Step by step solution

01

- Isolate the term with the variable

Divide both sides of the equation by C to isolate the term with the variable r.\[\frac{M}{C}=1+r\]
02

- Solve for r

Subtract 1 from both sides to solve for r.\[r=\frac{M}{C}-1\]

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Key Concepts

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

Variable Isolation
In algebra, variable isolation is an essential concept. It means rearranging an equation to get the unknown variable by itself on one side of the equation. We do this by using operations that simplify the equation step-by-step. For example, in the exercise, we started with the equation: \[ M = C(1 + r) \]The goal was to solve for the variable 'r.' To isolate 'r,' we first needed to get rid of the coefficient 'C' attached to the term containing 'r.' We did this by dividing both sides by 'C,' leading us to:\[ \frac{M}{C} = 1 + r \]Once the term containing 'r' was isolated, we could then proceed to the next steps to solve the equation.
Algebraic Manipulation
Algebraic manipulation is a set of techniques used to rearrange and simplify expressions. It involves operations like addition, subtraction, multiplication, and division, aimed at making the equation easier to work with.
In our case, after isolating the term containing 'r' by dividing both sides by 'C,' we performed another manipulation.
We had \[ \frac{M}{C} = 1 + r \] Next, we subtracted '1' from both sides to further isolate 'r.' This gave us:
\[ \frac{M}{C} - 1 = r \]
The key here was ensuring each operation maintained the equality. Knowing when and how to perform these steps is crucial in algebra and aids in breaking down complex problems into manageable parts.
Equation Solving
Solving equations is the process of finding the value of the variable that makes the equation true. Once we had isolated 'r' and performed the necessary algebraic manipulations, we arrived at:
\[ r = \frac{M}{C} - 1 \]
This is the solution for the variable 'r.' The process involved:
  • Isolating the term with 'r' by dividing both sides by 'C.'

  • Using algebraic manipulation to further isolate 'r' by subtracting 1 from both sides.

When solving equations, always check your work by substituting the solution back into the original equation to ensure both sides are equal. This verification step confirms that the solution is correct and satisfies the equation.

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