Question

At what temperature are the Fahrenheit and Celsius temperatures exactly equal?

Short answer

The temperatures are equal at -40 degrees.

Step-by-step solution

Understand the Relationship between Fahrenheit and Celsius

The relationship between Fahrenheit (\textdegree F) and Celsius (\textdegree C) is given by the formula: \[ F = \frac{9}{5}C + 32 \]

Set Fahrenheit Equal to Celsius

Since we need to find the temperature where Fahrenheit and Celsius are equal, set \( F = C \). So, the equation becomes: \[ C = \frac{9}{5}C + 32 \]

Solve for C

To solve for C, first subtract \( \frac{9}{5}C \) from both sides of the equation:\[ C - \frac{9}{5}C = 32 \]Combine the terms on the left side:\[ \frac{5}{5}C - \frac{9}{5}C = 32 \]\[ \frac{-4}{5}C = 32 \]Multiply both sides by \( -\frac{5}{4} \) to solve for C:\[ C = 32 \times -\frac{5}{4} \]\[ C = -40 \]

Key concepts

Fahrenheit and Celsius relationship

Understanding the relationship between Fahrenheit and Celsius is crucial when converting temperatures. The formula linking these two scales is: \[ F = \frac{9}{5}C + 32 \] This equation tells us that to convert a temperature from Celsius (\textdegree C) to Fahrenheit (\textdegree F), we multiply by 9/5 and then add 32.
This conversion is linear, meaning each degree change in Celsius results in a consistent change in Fahrenheit.
It's useful to know because it helps in various real-world scenarios like cooking or weather forecasting. Remembering this formula makes conversions straightforward and hassle-free.

Algebra in temperature equations

Using algebra in temperature equations involves solving for unknown temperatures.
In the exercise, we are tasked with finding at what temperature Fahrenheit and Celsius are equal.
We start with the relation: \[ F = \frac{9}{5}C + 32 \] To find the point where both temperatures are the same, we set \[ F = C \] Thus, our equation is reformed to: \[ C = \frac{9}{5}C + 32 \] Using algebra, we can isolate C (Celsius) and solve the equation:

• First, subtract \( \frac{9}{5}C \) from both sides: \[ C - \frac{9}{5}C = 32 \]
• Combine the terms: \[ \frac{5}{5}C - \frac{9}{5}C = 32 \]
• Simplify to: \( \frac{-4}{5}C = 32 \)
• To isolate C, multiply both sides by \( -\frac{5}{4} \)
• Get: \[ C = 32 \times -\frac{5}{4} \]
• Final answer: \[ C = -40 \]

This approach shows how algebra helps in solving temperature equations by systematically isolating the unknown variable.

Equal temperature point

The equal temperature point occurs when both Fahrenheit and Celsius readings are the same.
From our earlier calculations, we identified this point as \(-40\) degrees.
Here’s a straightforward way to understand this phenomenon:
At -40 degrees, neither the expansion nor compression of the temperature scales matters, as they perfectly align at this extreme point.
This unique convergence means: \[ -40 \textdegree C = -40 \textdegree F \]
It's an interesting and rare occurrence resulting directly from the algebraic alignment of the temperature conversion formula.
So, whenever asked about a temperature where both scales show the same value, you can confidently say it’s -40 degrees.
This concept highlights how mathematical principles help unravel even the most counterintuitive phenomena.