1. **Problem statement:** We have two perpendicular lines $l_1$ and $l_2$ intersecting at $(5,0)$. Line $l_2$ passes through $(0,7)$ and $(5,0)$, and line $l_1$ passes through $(0,-2)$ and $(5,0)$. (a) Find the gradient of line $l_1$. (b) Write the equation of line $l_1$ in the form $ax + by + d = 0$ with integers $a,b,d$ and $a > 0$. 2. **Formula for gradient:** The gradient (slope) $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$ is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Calculate gradient of $l_1$:** Using points $(0,-2)$ and $(5,0)$, $$m_{l_1} = \frac{0 - (-2)}{5 - 0} = \frac{2}{5}$$ 4. **Equation of line $l_1$:** Use point-slope form with point $(5,0)$ and slope $\frac{2}{5}$: $$y - 0 = \frac{2}{5}(x - 5)$$ Simplify: $$y = \frac{2}{5}x - 2$$ 5. **Rewrite in standard form $ax + by + d = 0$:** Multiply both sides by 5 to clear denominator: $$5y = 2x - 10$$ Bring all terms to one side: $$2x - 5y - 10 = 0$$ Here, $a=2$, $b=-5$, $d=-10$, and $a > 0$ as required. **Final answers:** (a) Gradient of $l_1$ is $\frac{2}{5}$. (b) Equation of $l_1$ is $2x - 5y - 10 = 0$.