Parallel Line Efed71

1. **State the problem:** We need to find the equation of a line that is parallel to the line $2x - 3y = -4$ and passes through the point $(-5, 3)$. 2. **Recall the formula and rules:** - Lines that are parallel have the same slope. - The given line is in standard form: $Ax + By = C$. - To find the slope, rewrite the given line in slope-intercept form $y = mx + b$. 3. **Find the slope of the given line:** Start with the given line: $$2x - 3y = -4$$ Solve for $y$: $$-3y = -2x - 4$$ $$y = \frac{-2x - 4}{-3} = \frac{2}{3}x + \frac{4}{3}$$ So, the slope $m$ of the given line is: $$m = \frac{2}{3}$$ 4. **Use the slope for the parallel line:** Since parallel lines have the same slope, the line we want also has slope: $$m = \frac{2}{3}$$ 5. **Use point-slope form to find the equation:** The point-slope form is: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1) = (-5, 3)$ and $m = \frac{2}{3}$. Substitute: $$y - 3 = \frac{2}{3}(x - (-5))$$ $$y - 3 = \frac{2}{3}(x + 5)$$ 6. **Simplify the equation:** Distribute: $$y - 3 = \frac{2}{3}x + \frac{2}{3} \times 5 = \frac{2}{3}x + \frac{10}{3}$$ Add 3 to both sides: $$y = \frac{2}{3}x + \frac{10}{3} + 3$$ Convert 3 to thirds: $$3 = \frac{9}{3}$$ So: $$y = \frac{2}{3}x + \frac{10}{3} + \frac{9}{3} = \frac{2}{3}x + \frac{19}{3}$$ 7. **Final answer:** The equation of the line parallel to $2x - 3y = -4$ and passing through $(-5, 3)$ is: $$y = \frac{2}{3}x + \frac{19}{3}$$