1. **Simplify each algebraic expression:** i) Simplify $7a + 3b - 5a + b - 6$ Combine like terms: $7a - 5a = 2a$, $3b + b = 4b$, so expression becomes $$2a + 4b - 6$$ ii) Simplify $m + 4mn - 2n$ No like terms to combine, so the expression remains $$m + 4mn - 2n$$ iii) Simplify $a + 2ab^2 - 3a^2b - ab^2 + ba^2$ Rearranging terms: $a + (2ab^2 - ab^2) + (-3a^2b + ba^2)$ Simplify like terms: $2ab^2 - ab^2 = ab^2$, $-3a^2b + a^2b = -2a^2b$, So final expression: $$a + ab^2 - 2a^2b$$ iv) Simplify $x^2 + 8x + xy + ax^2$ Combine like terms $x^2$ and $ax^2$: $1x^2 + ax^2 = (1 + a)x^2$, other terms remain, so $$ (1 + a)x^2 + 8x + xy$$ 2. **Express each as a single fraction:** i) Simplify $\frac{x}{5} + \frac{x}{2}$ Find common denominator 10: $\frac{2x}{10} + \frac{5x}{10} = \frac{7x}{10}$ ii) Simplify $\frac{4}{3x} - \frac{1}{4x}$ Common denominator $12x$: $\frac{16}{12x} - \frac{3}{12x} = \frac{13}{12x}$ iii) Simplify $\frac{x+1}{5} + \frac{x}{4}$ Common denominator 20: $\frac{4(x+1)}{20} + \frac{5x}{20} = \frac{4x + 4 + 5x}{20} = \frac{9x + 4}{20}$ iv) Simplify $x + \frac{2x-3}{5}$ Rewrite $x$ as $\frac{5x}{5}$: $\frac{5x}{5} + \frac{2x - 3}{5} = \frac{5x + 2x - 3}{5} = \frac{7x - 3}{5}$ 3. **Make the letter the subject:** i) Given $A = \frac{1}{2}(a + b)h$, solve for $h$ Multiply both sides by 2: $2A = (a + b)h$ Divide both sides by $(a + b)$: $$h = \frac{2A}{a + b}$$ ii) Given $A = a \sqrt{n^2 + r^2}$, solve for $r$ Divide both sides by $a$: $ \frac{A}{a} = \sqrt{n^2 + r^2}$ Square both sides: $\left(\frac{A}{a}\right)^2 = n^2 + r^2$ Isolate $r^2$: $r^2 = \left(\frac{A}{a}\right)^2 - n^2$ Take square root: $$r = \pm \sqrt{\left(\frac{A}{a}\right)^2 - n^2}$$ iii) Given $Y = \frac{x + 9}{x - 9}$, solve for $x$ Multiply both sides by $(x - 9)$: $Y(x - 9) = x + 9$ Expand left: $Yx - 9Y = x + 9$ Bring $x$ terms to one side: $Yx - x = 9 + 9Y$ Factor $x$: $x(Y - 1) = 9(1 + Y)$ Divide both sides by $(Y - 1)$: $$x = \frac{9(1 + Y)}{Y - 1}$$