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Chapter 2: Problem 72

Solve for the specified variable or expression. $$ \frac{3}{4} m-t=5 b \text { for } m $$

Short Answer

Expert verified
The solution is \( m = \frac{4}{3} (5b + t) \).

Step by step solution

01

Isolate the term with m

To solve for the variable \( m \), first isolate the term involving \( m \) on one side of the equation. Start by adding \( t \) to both sides of the equation to remove \( t \): \[ \frac{3}{4} m - t + t = 5b + t \] This simplifies to: \[ \frac{3}{4} m = 5b + t \]
02

Eliminate the fraction

Since \( m \) is still part of a fraction, eliminate the fraction by multiplying both sides of the equation by \( \frac{4}{3} \) to solve for \( m \): \[ \frac{4}{3} \cdot \frac{3}{4} m = (5b + t) \cdot \frac{4}{3} \] This simplifies to: \[ m = \frac{4}{3} (5b + t) \]

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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, isolating the variable of interest is a crucial first step. This is like cleaning the table off before starting a project. In the exercise, you want to solve for the variable \( m \). This means you should get \( m \) alone on one side of the equation. To do this, follow a systematic approach:
  • Look at the equation and identify the terms that involve the variable \( m \) and those that do not.
  • Use basic arithmetic operations, such as addition or subtraction, to move other terms to the opposite side of the equation from \( m \). In our example, moving \(- t\) to the right involves simply adding \( t \) to both sides.

Isolating the variable sets you up for easier algebraic manipulation and simplifies the solving process. Remember, keeping equations balanced is key, so whatever you do to one side, do to the other.
Algebraic Manipulation
Algebraic manipulation involves rearranging the equation to break it down into more manageable parts. Once you have isolated \( m \) to one side, you can focus on handling the equation effectively.Here's where algebraic manipulation comes into play:
  • Evaluate what operations are necessary to completely solve for \( m \). In the isolated form of \( m \), it's useful to recognize that the equation might still contain fractions or other operations that keep \( m \) from being fully isolated.
  • If there are coefficients or other variables mixed with \( m \), plan the steps to eliminate them from the side of the equation \( m \) resides.

Carefully implementing these manipulative strategies is crucial to simplify the expression further, and it鈥檚 this careful rearrangement that helps you arrive at the clean solution.
Eliminating Fractions
Fractions can make solving equations seem trickier than they are. Eliminating them simplifies the process. In the example, \( m \) is part of a fraction: \( \frac{3}{4}m \). The goal is to express \( m \) in terms of integers and variables rather than fractions.
  • Find the reciprocal of the fraction involving \( m \). Here, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
  • Multiply both sides of the equation by this reciprocal to cancel the fraction out. This makes "unwrapping" the fraction easier and \( m \) appears on its own.
  • This step should leave you with \( m \) completely isolated, turning a complex expression into something straightforward and clear.

Once any fraction surrounding \( m \) is eliminated, your equation is much cleaner and ready for interpretation. This increases clarity and reduces the potential for mistakes in calculations.

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Most popular questions from this chapter

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