1. Solve the equation $5x - 1 = 8x + 1$. Step 1: Subtract $5x$ from both sides: $$-1 = 3x + 1$$ Step 2: Subtract 1 from both sides: $$-2 = 3x$$ Step 3: Divide both sides by 3: $$x = \frac{-2}{3}$$ 2. Solve $3(x - 2) = 15$. Step 1: Expand left side: $$3x - 6 = 15$$ Step 2: Add 6 to both sides: $$3x = 21$$ Step 3: Divide both sides by 3: $$x = 7$$ 3. Solve $5(x + 6) = 20$. Step 1: Expand left side: $$5x + 30 = 20$$ Step 2: Subtract 30 from both sides: $$5x = -10$$ Step 3: Divide both sides by 5: $$x = -2$$ 4. Solve $2(x - 1) = \frac{5}{2}$. Step 1: Expand left side: $$2x - 2 = \frac{5}{2}$$ Step 2: Add 2 to both sides (write 2 as $\frac{4}{2}$): $$2x = \frac{5}{2} + \frac{4}{2} = \frac{9}{2}$$ Step 3: Divide both sides by 2: $$x = \frac{9}{4}$$ 5. Solve $5x - 3(x - 1) = 39$. Step 1: Expand: $$5x - 3x + 3 = 39$$ Step 2: Simplify: $$2x + 3 = 39$$ Step 3: Subtract 3: $$2x = 36$$ Step 4: Divide by 2: $$x = 18$$ 6. Solve $\frac{p - 1}{2} - \frac{p - 2}{3} = 1$. Step 1: Find common denominator 6: $$\frac{3(p - 1)}{6} - \frac{2(p - 2)}{6} = 1$$ Step 2: Combine: $$\frac{3p - 3 - 2p + 4}{6} = 1$$ Step 3: Simplify numerator: $$\frac{p + 1}{6} = 1$$ Step 4: Multiply both sides by 6: $$p + 1 = 6$$ Step 5: Subtract 1: $$p = 5$$ 7. Solve $8x - 2(3x - 8) = 24$. Step 1: Expand the bracket: $$8x - 6x + 16 = 24$$ Step 2: Simplify: $$2x + 16 = 24$$ Step 3: Subtract 16: $$2x = 8$$ Step 4: Divide by 2: $$x = 4$$ 8. Calculate $l m (m - n)$ for $l = -2$, $m = 3$, and $n = 7$. Step 1: Calculate $m - n$: $$3 - 7 = -4$$ Step 2: Multiply by $m$: $$3 \times (-4) = -12$$ Step 3: Multiply by $l$: $$-2 \times (-12) = 24$$ 9. Given $3$ is a root of $5x^2 - px - 18 = 0$, find $p$. Step 1: Substitute $x = 3$: $$5(3)^2 - p(3) - 18 = 0$$ Step 2: Calculate squares and multiply: $$5 \times 9 - 3p - 18 = 0$$ $$45 - 3p - 18 = 0$$ Step 3: Simplify: $$27 - 3p = 0$$ Step 4: Add $3p$ to both sides: $$27 = 3p$$ Step 5: Divide both sides by 3: $$p = 9$$ 10. Calculate $\frac{a(a - b)}{bc}$ for $a = 4$, $b = -2$, $c = 3$. Step 1: Calculate numerator $a(a - b)$: $$4(4 - (-2)) = 4(4 + 2) = 4 \times 6 = 24$$ Step 2: Calculate denominator: $$b \times c = -2 \times 3 = -6$$ Step 3: Compute fraction: $$\frac{24}{-6} = -4$$ 11. Evaluate given $a=2$, $b=-3$, $c=0$: (i) $4a - 2b + 3c$ Step 1: Calculate: $$4 \times 2 - 2 \times (-3) + 3 \times 0 = 8 + 6 + 0 = 14$$ (ii) $a^c$ Step 1: Calculate: $$2^0 = 1$$ 12. Simplify: (i) $3m - 2(m + l)$ Step 1: Expand: $$3m - 2m - 2l = m - 2l$$ (ii) $\frac{3}{x} - \frac{2}{y-2}$ (already simplified) (a) Simplify $\frac{x - 3}{3} + \frac{x + 2}{4}$. Step 1: Find common denominator 12: $$\frac{4(x - 3)}{12} + \frac{3(x + 2)}{12}$$ Step 2: Expand numerators: $$\frac{4x - 12 + 3x + 6}{12} = \frac{7x - 6}{12}$$ Final answers: 1. $x = \frac{-2}{3}$ 2. $x = 7$ 3. $x = -2$ 4. $x = \frac{9}{4}$ 5. $x = 18$ 6. $p = 5$ 7. $x = 4$ 8. $24$ 9. $p = 9$ 10. $-4$ 11(i). $14$ 11(ii). $1$ 12(i). $m - 2l$ 12(ii). $\frac{3}{x} - \frac{2}{y-2}$ 12(a). $\frac{7x - 6}{12}$