To find the equation of the line that passes through the point \((-3, 5)\) and is parallel to the line defined by the equation \(-4x = 5y + 3\), we first need to determine the slope of the given line.
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Rearrange the equation to slope-intercept form (y = mx + b): \[ -4x = 5y + 3 \] \[ 5y = -4x - 3 \] \[ y = -\frac{4}{5}x - \frac{3}{5} \] From this, we can see that the slope (m) is \(-\frac{4}{5}\).
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Since parallel lines have the same slope, the slope of our new line will also be \(-\frac{4}{5}\).
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We use the point-slope form of the equation of a line to find the equation of the line that passes through \((-3, 5)\): \[ y - y_1 = m(x - x_1) \] Plugging in the values: \[ y - 5 = -\frac{4}{5}(x + 3) \]
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Simplifying this equation: \[ y - 5 = -\frac{4}{5}x - \frac{12}{5} \] \[ y = -\frac{4}{5}x - \frac{12}{5} + 5 \] Since \(5 = \frac{25}{5}\): \[ y = -\frac{4}{5}x - \frac{12}{5} + \frac{25}{5} \] \[ y = -\frac{4}{5}x + \frac{13}{5} \]
Thus, the slope-intercept form of the line is: \[ y = -\frac{4}{5}x + \frac{13}{5} \]
- Now, to express the equation in standard form \(Ax + By = C\):
- Multiply the entire equation by 5 to eliminate the fractions: \[ 5y = -4x + 13 \] Rearranging gives: \[ 4x + 5y = 13 \]
Therefore, the final equations of the line are:
Slope-Intercept Form:
\[
y = -\frac{4}{5}x + \frac{13}{5}
\]
Standard Form:
\[
4x + 5y = 13
\]