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