An affine function \( y = y(x) \) takes the value \( 81.5 \) when \( x = 13 \), and the value \( 96.7 \) when \( x = 20.6 \). What is the correspondence rule?
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\( y = 0.5x + 75 \)
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\( y = 2x + 55.5 \)
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\( y = 0.5x + 55 \)
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\( y = 2x + 75 \)
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\( y = 0.5x + 81.5 \)
Try to solve it before checking the answer.
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\( y = 2x + 55.5 \)
An affine function has the following form:
\[ \boldsymbol{(1)} \hspace{2em} y = ax + b \]Replacing the values of \( x = 13 \) and \( y = 81.5 \), and the values of \( x = 20.6 \) and \( y = 96.7 \), into equation (1), we obtain the system:
\[ \begin{aligned} &\boldsymbol{(2)} \hspace{2em} 81.5 = 13a + b \\[1em] &\boldsymbol{(3)} \hspace{2em} 96.7 = 20.6 a + b \end{aligned} \]subtracting \((2)\) from \((3)\): \[ 15.2 = 7.6a \Rightarrow a = \frac{15.2}{7.6} = \boldsymbol{2} \]
Replacing \( a=2 \) in the equation (2):
\[ 81.5 = 13(2) + b \Rightarrow b = 55.5 \]Hence,
\[ \boldsymbol{ y = 2x + 55.5 } \]