To find the equation of the line that passes through the points (2, –1) and (8, 4), we will first calculate the slope (m) of the line and then use the point-slope form of the equation.
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Calculate the slope (m): The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] With points (2, -1) \((x_1, y_1)\) and (8, 4) \((x_2, y_2)\): \[ m = \frac{4 - (-1)}{8 - 2} = \frac{4 + 1}{8 - 2} = \frac{5}{6} \]
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Write the equation in point-slope form: The point-slope form of a line is: \[ y - y_1 = m(x - x_1) \] Using point (2, -1): \[ y + 1 = \frac{5}{6}(x - 2) \]
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Rewrite the equation in standard form: To convert \(y + 1 = \frac{5}{6}(x - 2)\) into standard form, we start by eliminating the fraction. Multiply both sides by 6: \[ 6(y + 1) = 5(x - 2) \] Distributing both sides: \[ 6y + 6 = 5x - 10 \] Now rearranging to standard form \(Ax + By = C\): \[ -5x + 6y = -16 \] To make the coefficient of \(x\) positive, multiply through by -1: \[ 5x - 6y = 16 \]
So, the final standard form of the equation is: \[ 5x - 6y = 16 \]
From the options you provided, the closest form is:
- y + 1 = \frac{5}{6}(x - 2); -5x + 6y = -16
This indicates that the point-slope form equation is correct, but the standard form given in the options is not in the same sign.
Thus, the correct equation in point-slope form is: \[ y + 1 = \frac{5}{6}(x - 2) \]
And the correct standard form is: \[ 5x - 6y = 16 \]