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Chapter 2: Problem 37

Solve \(8 A+20 B=100\) for \(A\).

Short Answer

Expert verified
A = 12.5 - 2.5B

Step by step solution

01

- Isolate terms with A

Start by isolating the terms with the variable A. Subtract 20B from both sides of the equation:\[ 8A + 20B - 20B = 100 - 20B \]This simplifies to:\[ 8A = 100 - 20B \]
02

- Solve for A

Now, get A by itself by dividing both sides of the equation by 8:\[ A = \frac{100 - 20B}{8} \]Simplify the expression if possible:\[ A = \frac{100}{8} - \frac{20B}{8} \]Further simplifying:\[ A = 12.5 - 2.5B \]

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Key Concepts

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

Isolating Variables
When solving linear equations, the goal is often to isolate the target variable on one side of the equation. This involves moving all other terms to the opposite side. In our example, we want to solve for \(A\) in the equation \(8A + 20B = 100\). Start by isolating the term with \(A\). Subtract \(20B\) from both sides:
\[ 8A + 20B - 20B = 100 - 20B \]
This simplifies to:
\[ 8A = 100 - 20B \]
Notice how we shifted the term \(20B\) from the left side to the right by subtracting it from both sides. This process of moving terms is known as isolating the variable.
Simplifying Expressions
Once the variable is isolated on one side of the equation, the next step is to simplify the expression, making it easier to solve. If we look at our equation:
\[ 8A = 100 - 20B \]
The expression \(100 - 20B\) is already simplified. However, we need to isolate \(A\). To do this, divide both sides of the equation by 8:
\[ A = \frac{100 - 20B}{8} \]
This makes the equation easier to manage. Simplifying the fractions further, we get:
\[ A = \frac{100}{8} - \frac{20B}{8} \]
Resulting in:
\[ A = 12.5 - 2.5B \]
Simplifying ensures that the final form is clear and easy to understand.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to rearrange and solve equations. It requires understanding properties of equality and operations. Let's review our equation after isolating and simplifying:
\[ A = 12.5 - 2.5B \]
Here, we used both subtraction and division to achieve our final expression.
  • First, subtraction helped us move terms (\(20B\)).
  • Then, division allowed us to isolate \(A\) by dividing all terms by 8.

Mastering algebraic manipulation means becoming comfortable with these operations and knowing when to apply them. Practice is key! Remember to always perform the same operation on both sides of the equation to maintain balance and correctness.

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Most popular questions from this chapter

If the population of a city includes 75,000 people who are not immune to measles, find the minimum number of people who will get measles if they are exposed to the virus. Before measles immunization was available, nearly everyone in the U.S. got measles. ... Measles is one of the most infectious diseases in the world and is frequently imported into the U.S. ... More than 90 percent of people who are not immune will get measles if they are exposed to the virus. (Source: www.cdc.gov, Aug. 2010)

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