To find the equation of the linear function in the form \(y = mx + b\), we first need to determine the slope \(m\) using the two data points given: \((11, 58)\) and \((20, 103)\).
The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Plugging in our points:
- \(x_1 = 11\), \(y_1 = 58\)
- \(x_2 = 20\), \(y_2 = 103\)
Calculating the slope:
\[ m = \frac{103 - 58}{20 - 11} = \frac{45}{9} = 5 \]
Now that we have the slope \(m = 5\), we can use one of the points to find the y-intercept \(b\). We'll use the point \((11, 58)\).
Substituting \(x = 11\), \(y = 58\), and \(m = 5\) into the equation \(y = mx + b\):
\[ 58 = 5(11) + b \]
Calculating:
\[ 58 = 55 + b \]
To isolate \(b\), subtract 55 from both sides:
\[ b = 58 - 55 = 3 \]
Now that we have \(m\) and \(b\), we can write the equation of the linear function:
\[ y = 5x + 3 \]
Thus, the equation that models the relationship shown in the table is:
\[ \boxed{y = 5x + 3} \]
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