To find the equation of the line that passes through the point \((5, -3)\) and is parallel to the line given by the equation \(4x + 5y = 15\), we first need to determine the slope of the given line.
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Convert the given line equation to slope-intercept form (y = mx + b): Starting with: \[ 4x + 5y = 15 \] We can isolate \(y\): \[ 5y = -4x + 15 \] \[ y = -\frac{4}{5}x + 3 \] This tells us that the slope \(m\) of the given line is \(-\frac{4}{5}\).
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Since parallel lines have the same slope, the slope of the line we want to find will also be \(-\frac{4}{5}\).
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Now, use the point-slope form of the line equation: The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1) = (5, -3)\) and \(m = -\frac{4}{5}\): \[ y - (-3) = -\frac{4}{5}(x - 5) \] Simplifying this gives: \[ y + 3 = -\frac{4}{5}x + 4 \] \[ y = -\frac{4}{5}x + 4 - 3 \] \[ y = -\frac{4}{5}x + 1 \]
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Convert this to standard form (Ax + By = C): Start with the slope-intercept form: \[ y = -\frac{4}{5}x + 1 \] To eliminate the fraction, multiply the entire equation by 5: \[ 5y = -4x + 5 \] Rearranging to standard form yields: \[ 4x + 5y = 5 \]
Therefore, the equation of the line that passes through the point \((5, -3)\) and is parallel to the line \(4x + 5y = 15\) is: \[ \boxed{4x + 5y = 5} \]
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