1. **Problem:** Find the equation of a straight line with gradient $m$ passing through a point $(x_1, y_1)$. 2. **Formula:** The point-slope form of a line is given by: $$y - y_1 = m(x - x_1)$$ This formula is used to find the equation of a line when you know the gradient and a point on the line. 3. **Important rules:** - The gradient $m$ is the slope of the line. - $(x_1, y_1)$ is a point on the line. - After substituting values, simplify to get the equation in slope-intercept form $y = mx + c$. 4. **Solve part (a):** Gradient $m=2$, point $(3,2)$ $$y - 2 = 2(x - 3)$$ $$y - 2 = 2x - 6$$ $$y = 2x - 6 + 2$$ $$y = 2x - 4$$ 5. **Solve part (b):** Gradient $m=-1$, point $(2,1)$ $$y - 1 = -1(x - 2)$$ $$y - 1 = -x + 2$$ $$y = -x + 2 + 1$$ $$y = -x + 3$$ 6. **Solve part (c):** Gradient $m=3$, point $(-2,0)$ $$y - 0 = 3(x + 2)$$ $$y = 3x + 6$$ 7. **Solve part (d):** Gradient $m=-3$, point $(1,-2)$ $$y + 2 = -3(x - 1)$$ $$y + 2 = -3x + 3$$ $$y = -3x + 3 - 2$$ $$y = -3x + 1$$ 8. **Solve part (e):** Gradient $m=3$, passes through origin $(0,0)$ $$y - 0 = 3(x - 0)$$ $$y = 3x$$ 9. **Solve part (f):** Gradient $m=4$, point $(-1,-2)$ $$y + 2 = 4(x + 1)$$ $$y + 2 = 4x + 4$$ $$y = 4x + 4 - 2$$ $$y = 4x + 2$$ 10. **Solve part (g):** Gradient $m=4$, point $(2,8)$ $$y - 8 = 4(x - 2)$$ $$y - 8 = 4x - 8$$ $$y = 4x - 8 + 8$$ $$y = 4x$$ 11. **Solve part (h):** Gradient $m=-2$, point $(-6,4)$ $$y - 4 = -2(x + 6)$$ $$y - 4 = -2x - 12$$ $$y = -2x - 12 + 4$$ $$y = -2x - 8$$ 12. **Solve part (i):** Gradient $m=2$, midpoint of $(1,3)$ and $(5,5)$ Midpoint coordinates: $$x_m = \frac{1 + 5}{2} = 3, \quad y_m = \frac{3 + 5}{2} = 4$$ Equation: $$y - 4 = 2(x - 3)$$ $$y - 4 = 2x - 6$$ $$y = 2x - 6 + 4$$ $$y = 2x - 2$$ 13. **Solve part (j):** Gradient $m=\frac{1}{2}$, midpoint of $(0,0)$ and $(-6,4)$ Midpoint coordinates: $$x_m = \frac{0 - 6}{2} = -3, \quad y_m = \frac{0 + 4}{2} = 2$$ Equation: $$y - 2 = \frac{1}{2}(x + 3)$$ $$y - 2 = \frac{1}{2}x + \frac{3}{2}$$ $$y = \frac{1}{2}x + \frac{3}{2} + 2$$ $$y = \frac{1}{2}x + \frac{7}{2}$$ **Final answers:** (a) $y = 2x - 4$ (b) $y = -x + 3$ (c) $y = 3x + 6$ (d) $y = -3x + 1$ (e) $y = 3x$ (f) $y = 4x + 2$ (g) $y = 4x$ (h) $y = -2x - 8$ (i) $y = 2x - 2$ (j) $y = \frac{1}{2}x + \frac{7}{2}$