1. **Problem 2:** Find the gradient for each line and express the equation in the form $ax+by+c=0$. 2. **Problem 3:** Find the equation of the line through point $(2,5)$ with gradient $3$ and express it in the form $ax+by+c=0$. 3. **Problem 6:** Find the equation of the line through point $(-3,4)$ with gradient $-\frac{2}{5}$ and express it in the form $ax+by+c=0$. --- ### Step-by-step solutions: **2) Gradient given, find equation in $ax+by+c=0$ form:** Since the problem states "Find the gradient for each line" but no specific lines are given, we proceed to problems 3 and 6 which involve gradients and points. --- **3) Line through $(2,5)$ with gradient $3$:** 1. Use point-gradient form: $$y - y_1 = m(x - x_1)$$ 2. Substitute $m=3$, $x_1=2$, $y_1=5$: $$y - 5 = 3(x - 2)$$ 3. Expand right side: $$y - 5 = 3x - 6$$ 4. Bring all terms to one side: $$y - 5 - 3x + 6 = 0$$ 5. Simplify constants: $$-3x + y + 1 = 0$$ 6. Multiply by $-1$ to make $a$ positive: $$3x - y - 1 = 0$$ This is the equation in the form $ax+by+c=0$. --- **6) Line through $(-3,4)$ with gradient $-\frac{2}{5}$:** 1. Use point-gradient form: $$y - y_1 = m(x - x_1)$$ 2. Substitute $m = -\frac{2}{5}$, $x_1 = -3$, $y_1 = 4$: $$y - 4 = -\frac{2}{5}(x + 3)$$ 3. Expand right side: $$y - 4 = -\frac{2}{5}x - \frac{6}{5}$$ 4. Bring all terms to one side: $$y - 4 + \frac{2}{5}x + \frac{6}{5} = 0$$ 5. Combine constants: $$y + \frac{2}{5}x - \frac{20}{5} + \frac{6}{5} = 0$$ $$y + \frac{2}{5}x - \frac{14}{5} = 0$$ 6. Multiply entire equation by $5$ to clear denominators: $$5y + 2x - 14 = 0$$ 7. Rearrange to standard form: $$2x + 5y - 14 = 0$$ --- **Final answers:** - Problem 3: $$3x - y - 1 = 0$$ - Problem 6: $$2x + 5y - 14 = 0$$