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Chapter 5: Problem 76

Solve for the specified variable or expression. $$ 2 g=c h+d h \text { for } h $$

Short Answer

Expert verified
\( h = \frac{2g}{c + d} \)

Step by step solution

01

Factor the right-hand side

First, notice that we can factor the right side of the equation. Both terms on the right-hand side of the equation have a common factor of \( h \). Therefore, we can rewrite the expression as: \( 2g = h(c + d) \).
02

Isolate the variable \( h \)

To solve for \( h \), we need to isolate it on one side of the equation. We do this by dividing both sides of the equation by \( (c + d) \). Thus, the equation becomes \( h = \frac{2g}{c + d} \).

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Key Concepts

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

Factoring
In algebra, factoring is a process where you express an expression as a product of its factors. When dealing with equations, this can often simplify solving them by making it easier to isolate terms.
Consider the equation from our original exercise:
\[2g = ch + dh\]
Both terms on the right-hand side contain the variable \(h\). This means \(h\) is a common factor. By factoring out \(h\), we can rewrite the expression more compactly as:
\[2g = h(c + d)\]
  • Factoring simplifies complex expressions.
  • Identifying common factors is the key step in factoring.
Recognizing common factors and rewriting equations in factored form often unveils the solution path more clearly.
Isolating Variables
Isolating a variable is a fundamental technique in solving algebraic equations. The aim is to get the variable we are solving for, in this case \(h\), on one side of the equation on its own.
After factoring, the equation becomes:\[ 2g = h(c + d) \]To isolate \(h\):
  • Divide both sides of the equation by the same expression \((c + d)\) to keep the equation balanced.
Thus, your equation turns into:\[h = \frac{2g}{c+d}\]This step ensures that \(h\) stands alone, making it clear what value \(h\) represents in terms of the other variables and constants.
Algebraic Manipulation
Algebraic manipulation encompasses a variety of strategies used to rearrange and simplify equations or expressions in order to solve for a particular variable or to make the expression easier to work with.
Some common techniques include:
  • Adding or subtracting the same quantity from both sides of the equation.
  • Multiplying or dividing both sides by the same nonzero number.
  • Factoring and distributing.
In our exercise, these principles come into play as we perform both factoring and division to isolate the variable \(h\). Understanding these techniques:
  • Makes solving equations systematic.
  • Ensures you maintain equality throughout the steps.
Algebraic manipulation is an essential skill that enhances problem-solving capabilities in not just mathematics but also in various fields requiring analytical thinking.

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