Chapter 2: Introduction to Linear Polynomials
Chapter 2: Introduction to Linear Polynomials
Exercise
Question
Question: 3
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Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by °C = a °F + b. Find a and b, given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit.
Hint: When °C = 0, °F = 32 and when °C = 100, °F = 212. Use this information to find a and b, and thus, the linear relationship between °C and °F.)
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Answer
1
Use the relation °C = a °F + b and the two given temperature pairs to form two equations in a and b.
\text{$0 = a × 32 + b$ $\quad$ ${color:#6D28D9}[Equation 1]{/color}$ $\quad$ ${color:#625B5F}[°C = 0, °F = 32]{/color}$}
\text{$100 = a × 212 + b$ $\quad$ ${color:#6D28D9}[Equation 2]{/color}$ $\quad$ ${color:#625B5F}[°C = 100, °F = 212]{/color}$}
2
Subtract the first equation from the second equation to find a.
$100 - 0 = (212a + b) - (32a + b)$
⇒
$100 = 212a + b - 32a - b$
⇒
$100 = 180a$
⇒
\text{ $a = \dfrac{100}{180}$}
⇒
\text{ $a = \dfrac{5}{9}$}
3
Substitute the value of a into the first equation to find b.
\text{$0 = {color:#2E2EFF}\dfrac{5}{9}{/color} × 32 + b$}
⇒
\text{ $b = -\dfrac{5 × 32}{9}$}
⇒
\text{ $b = -\dfrac{160}{9}$}
4
Write the values of a and b.
\textbf{Answer: $a = \dfrac{5}{9}$, $b = -\dfrac{160}{9}$}