To find the linear equation that models the relationship between \(x\) and \(y\) in the given table, we need to determine the slope and y-intercept.
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Calculate the slope (m): Using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the first two points (52, 39) and (61, 48): \[ m = \frac{48 - 39}{61 - 52} = \frac{9}{9} = 1 \]
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Find the y-intercept (b): We can use the slope-intercept form of the equation: \[ y = mx + b \]
We know \(m = 1\). Now we can use one of the points to find \(b\). Using the point (52, 39): \[ 39 = 1(52) + b \] \[ b = 39 - 52 = -13 \]
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Write the equation: Now that we have the slope and y-intercept, we can write the equation: \[ y = 1x - 13 \]
Thus, the linear equation is: \[ y = x - 13 \]