To determine the temperature at which Celsius and Fahrenheit are equal, we need to find the point where the temperature reading in Celsius (C) and Fahrenheit (F) are the same.
We start with the formula that converts Celsius to Fahrenheit:
[tex]\[ F = \frac{9}{5}C + 32 \][/tex]
Since we're looking for the temperature at which [tex]\( C \)[/tex] and [tex]\( F \)[/tex] are equal, we set [tex]\( F = C \)[/tex]. Substituting [tex]\( C \)[/tex] for [tex]\( F \)[/tex] in the formula, we get:
[tex]\[ C = \frac{9}{5}C + 32 \][/tex]
Next, we'll solve for [tex]\( C \)[/tex]. Begin by isolating the terms involving [tex]\( C \)[/tex] on one side of the equation:
[tex]\[ C - \frac{9}{5}C = 32 \][/tex]
To combine the [tex]\( C \)[/tex] terms, factor out [tex]\( C \)[/tex]:
[tex]\[ C \left(1 - \frac{9}{5}\right) = 32 \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ 1 - \frac{9}{5} = \frac{5}{5} - \frac{9}{5} = \frac{5 - 9}{5} = -\frac{4}{5} \][/tex]
So the equation now is:
[tex]\[ C \left(-\frac{4}{5}\right) = 32 \][/tex]
To solve for [tex]\( C \)[/tex], multiply both sides of the equation by the reciprocal of [tex]\( -\frac{4}{5} \)[/tex], which is [tex]\( -\frac{5}{4} \)[/tex]:
[tex]\[ C = 32 \times -\frac{5}{4} \][/tex]
Perform the multiplication:
[tex]\[ C = 32 \times -\frac{5}{4} = 32 \times -1.25 = -40 \][/tex]
Therefore, the temperature at which Celsius and Fahrenheit are equal is:
[tex]\[ -40 \,\text{degrees} \][/tex]
Thus, Celsius and Fahrenheit are equal at [tex]\( -40 \)[/tex] degrees.
