1. **Stating the problem:** Find the equations of the lines passing through given points and express them in slope-intercept form $y = mx + b$. 2. **Line R through points $(-2,1)$ and $(4,4)$:** - Use the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{4 - (-2)} = \frac{3}{6} = \frac{1}{2}$$ - Using point-slope form with point $(-2,1)$: $$1 = m \cdot (-2) + b$$ - Substitute $m = \frac{1}{2}$: $$1 = \frac{1}{2} \cdot (-2) + b = -1 + b$$ - Solve for $b$: $$b = 1 + 1 = 2$$ - Equation of line R: $$y = \frac{1}{2}x + 2$$ 3. **Line N through point $(2,-5)$ with slope $m = \frac{6}{5}$:** - Use point-slope form: $$-5 = \frac{6}{5} \cdot 2 + b$$ - Calculate: $$-5 = \frac{12}{5} + b$$ - Solve for $b$: $$b = -5 - \frac{12}{5} = -\frac{25}{5} - \frac{12}{5} = -\frac{37}{5}$$ - Equation of line N: $$y = \frac{6}{5}x - \frac{37}{5}$$ 4. **Summary:** - Line R: $$y = \frac{1}{2}x + 2$$ - Line N: $$y = \frac{6}{5}x - \frac{37}{5}$$ Note: The original problem's calculations for $m$ and $b$ for line R differ; here we used the standard slope formula for clarity and correctness.