Bot GPT 4-o mini To find the equation of the linear function in the form \(y = mx + b\), we need to determine the slope \(m\) and the y-intercept \(b\).
- 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: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((6, 16)\) and \((15, 43)\):
- \(x_1 = 6\), \(y_1 = 16\)
- \(x_2 = 15\), \(y_2 = 43\)
Substituting these values into the slope formula: \[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]
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Use the slope and one of the points to find the y-intercept \(b\): We can use the point-slope form \(y = mx + b\). Let's use the point \((6, 16)\): \[ 16 = 3(6) + b \] Calculating: \[ 16 = 18 + b \] \[ b = 16 - 18 = -2 \]
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Write the equation: Now that we have both \(m\) and \(b\), we can write the equation of the line: \[ y = 3x - 2 \]
Thus, the equation of the linear function that models the relationship shown in the table is: \[ \boxed{y = 3x - 2} \]
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