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Chapter 1: Problem 25

Solve. \(C=\frac{5}{9}(F-32),\) for \(F\) (a temperature formula)

Short Answer

Expert verified
F = \frac{9}{5}C + 32

Step by step solution

01

- Isolate the Variable Term

Start by isolating the term that contains the variable we are solving for. The given formula is \[C = \frac{5}{9}(F - 32)\]. Multiply both sides of the equation by \(\frac{9}{5}\) to cancel out the fraction on the right-hand side.\[\frac{9}{5}C = F - 32\]
02

- Solve for the Variable

Add 32 to both sides to solve for \(F\):\[ F = \frac{9}{5}C + 32\]

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Key Concepts

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

Solving Equations
When faced with an equation, the goal is often to find the value of an unknown variable. This process is known as solving equations.
The essence of solving equations involves
  • Identifying the variable you need to solve for
  • Performing operations until the variable stands alone on one side of the equation
Every step must maintain the balance of the equation so the equality is preserved.
You will often find yourself performing operations like addition, subtraction, multiplication, and division to achieve this goal.
Isolation of Variables
Isolation of variables is a key step in solving equations. This involves moving all terms involving the variable to one side of the equation.
Take the exercise given: \[ C = \frac{5}{9}(F - 32) \] We aim to solve for F. First, isolate terms with F by multiplying both sides by \[\frac{9}{5}\]. This yields:
\[\frac{9}{5}C = F - 32\]
Notice how the fraction is eliminated on the right.
The next step is to get F completely isolated by adding 32 to both sides:
\
Algebraic Manipulation
Algebraic manipulation involves the application of arithmetic operations to rearrange equations. This skill is essential for isolation of variables and solving equations.
Our exercise demonstrates this well. Here are the key operations used:
  • Multiplication: We multiplied both sides by \[\frac{9}{5}\] to clear the fraction.
  • Addition: We added 32 to both sides to isolate F.
Combining these steps:
\[\frac{9}{5}C = F - 32\] with
\[\frac{9}{5}C + 32 = F\]
shows how manipulating terms can make solving for F straightforward.
Temperature Scales
Understanding temperature scales is crucial for problems like these. The most common scales are Celsius (C) and Fahrenheit (F).
The formula \[C = \frac{5}{9}(F - 32)\] converts Fahrenheit to Celsius. By isolating F, we obtained the reverse:
\[F = \frac{9}{5}C + 32\]
Knowing these conversions helps in various practical scenarios, such as weather forecasting, cooking, and scientific experiments. Remember:
  • To convert Fahrenheit to Celsius: use \[C = \frac{5}{9}(F - 32)\]
  • To convert Celsius to Fahrenheit: use \[F = \frac{9}{5}C + 32\]
Understanding these conversions is not just academic; it has real-world applications too.

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Most popular questions from this chapter

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{4^{3}-(7-4)^{2}}{3^{2}-7} $$

Find an equivalent expression by factoring. $$ x y+x $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -(-3)^{2} $$

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -6.08,-5,0,1, \sqrt{17}, \frac{99}{2} $$

Find an equivalent expression by factoring. $$ 5 x+50 $$
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