Parallel Lines 027489

1. **State the problem:** Find the equations of lines passing through the point $(10,-2)$ parallel to the given lines $4x-2y+1=0$ and $x+5y-10=0$. 2. **Find the slope of each given line:** - For line a: $4x - 2y + 1 = 0$ Rewrite in slope-intercept form $y = mx + b$: $$4x - 2y + 1 = 0 \implies -2y = -4x - 1 \implies y = \frac{-4x - 1}{-2} = 2x + \frac{1}{2}$$ So, slope $m_a = 2$. - For line b: $x + 5y - 10 = 0$ Rewrite: $$5y = -x + 10 \implies y = -\frac{1}{5}x + 2$$ So, slope $m_b = -\frac{1}{5}$. 3. **Use point-slope form for parallel lines:** The equation of a line with slope $m$ passing through $(x_1,y_1)$ is: $$y - y_1 = m(x - x_1)$$ 4. **Write equations for each line:** - For line a (slope $2$): $$y - (-2) = 2(x - 10)$$ $$y + 2 = 2x - 20$$ $$y = 2x - 22$$ - For line b (slope $-\frac{1}{5}$): $$y - (-2) = -\frac{1}{5}(x - 10)$$ $$y + 2 = -\frac{1}{5}x + 2$$ $$y = -\frac{1}{5}x + 2 - 2$$ $$y = -\frac{1}{5}x$$ 5. **Final answers:** - Line parallel to $4x - 2y + 1 = 0$ through $(10,-2)$ is: $$y = 2x - 22$$ - Line parallel to $x + 5y - 10 = 0$ through $(10,-2)$ is: $$y = -\frac{1}{5}x$$