Answer
(a)
$C=-30°$
$F=-22°$
$C=-20°$
$F=-4°$
$C=-10°$
$F=14°$
$C=0°$
$F=32°$
$F=50°$
$C=10°$
$F=68°$
$C=20°$
$F=86°$
$C=30°$
(b) The scales agree at $-40°$
Work Step by Step
$F = \frac{9}{5}C+32$
(a)
$C=-30°$
$F=\frac{9}{5}\times (-30) + 32=-54+32=-22°$
$C=-20°$
$F=\frac{9}{5}\times (-20) + 32=-36+32=-4°$
$C=-10°$
$F=\frac{9}{5}\times (-10) + 32=-18+32=14°$
$C=0°$
$F=\frac{9}{5}\times 0 + 32=32°$
To make our calculation easier while converting Fahrenheit to Celsius, let's rewrite the equation. Solve it for $C$ instead of $F$:
$\frac{9}{5}C=F-32$
$C=(F-32)\frac{5}{9}$
$F=50°$
$C=(50-32)\frac{5}{9}=18\times \frac{5}{9}=10°$
$F=68°$
$C=(68-32)\frac{5}{9}=36\times \frac{5}{9}=20°$
$F=86°$
$C=(86-32)\frac{5}{9}=54\times \frac{5}{9}=30°$
(b) For the scales to be the same, values of their equations has to be the same, so we can write:
$\frac{9}{5}C+32=(F-32)\frac{5}{9}$
Also, we can conclude that $F=C=a$
$\frac{9}{5}a+32=(a-32)\frac{5}{9}$
$\frac{9}{5}a+32=\frac{5}{9}a-\frac{160}{9}$ //Multiply by $45$
$81a+1440=25a-800$
$56a=-2240$
$a=-40$
The scales agree at $-40°$
