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Chapter 2: Problem 91

Solve each formula for the specified variable. $$ P=2(a+b) \text { for } a $$

Short Answer

Expert verified
a = \frac{P}{2} - b

Step by step solution

01

Original Equation

Start with the given equation: \[ P = 2(a + b) \]
02

Divide Both Sides by 2

To isolate the term with the variable \(a\), divide both sides of the equation by 2:\[ \frac{P}{2} = \frac{2(a + b)}{2} \]The equation simplifies to:\[ \frac{P}{2} = a + b \]
03

Isolate the Variable \(a\)

Subtract \(b\) from both sides to solve the equation for \(a\):\[ \frac{P}{2} - b = a + b - b \]This simplifies to:\[ \frac{P}{2} - b = a \]
04

Final Answer

The equation is now solved for the variable \(a\):\[ a = \frac{P}{2} - b \]

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Key Concepts

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

isolating variables
Isolating variables is often the first major step in solving an equation. This process involves moving everything except the variable of interest to the other side of the equation.
In our exercise, we isolate the variable \(a\) to solve for it.
We start with the equation:
\[P = 2(a + b)\].
Here, we see that \(a\) is inside a parenthesis and multiplied by 2.
To isolate \(a\):
  • First, divide both sides by 2 to get rid of the multiplication:
    \(\frac{P}{2} = a + b\).
  • Next, we need to get \(a\) by itself. Since \(b\) is added to \(a\), we subtract \(b\) from both sides:
    \( a = \frac{P}{2} - b \).
Yay! \(a\) is now alone, and we have our solution.
algebraic manipulation
Algebraic manipulation refers to the techniques used to rearrange and simplify equations. To solve equations effectively, you need to be comfortable with a few key operations:
  • Addition / Subtraction: You can add or subtract the same number from both sides of an equation without changing the equality.
  • Multiplication / Division: Likewise, multiplying or dividing both sides by the same non-zero number will not change the equality.
  • For our example, we use division to simplify the given equation:

Starting with:
\[ P = 2(a + b) \],
we divide both sides by 2:
\[ \frac{P}{2} = a + b \].
Then, we use subtraction to move \(b\) to the other side:
\[ \frac{P}{2} - b = a \].
These steps simplify the equation, leaving \(a\) by itself.
linear equations
Linear equations have terms raised to the power of 1 and graph as straight lines. They typically appear in the form \(Ax + B = C\).
In our problem, the equation \( P = 2(a + b) \) is linear because it can be rearranged into the simpler linear form. Let's deconstruct:
  • First, understand that a linear equation involves operations of addition, subtraction, multiplication, and division.
  • For example, \( 2(a + b) \) represents a multiplication: a classic trait of linear equations.

To solve for \(a\) from our exercise:
We recognize \(a\) and \(b\) are terms added together, multiplied by 2, and equal to \(P\). By isolating \(a\), we keep the equation linear throughout our steps:
\[ a = \frac{P}{2} - b \].
This final form reflects a linear relationship where \(a\) is solved in terms of\(P\) and \(b\). Easy, right?

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Most popular questions from this chapter

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