1. Simplify the expressions: 1.a) Simplify $y^2 + 2y - 5y^2 - 1 - 2 + y$ Combine like terms: $$y^2 - 5y^2 + 2y + y - 1 - 2 = (1 - 5)y^2 + (2 + 1)y - 3 = -4y^2 + 3y - 3$$ 1.b) Simplify $4a + 2n + 5z + 6 + n - 5a$ Group like terms: $$(4a - 5a) + (2n + n) + 5z + 6 = -1a + 3n + 5z + 6 = -a + 3n + 5z + 6$$ 2. Expand the brackets: 2.a) $4(7r - 7s + 10) = 4 \times 7r - 4 \times 7s + 4 \times 10 = 28r - 28s + 40$ 2.b) $5(4s - 3t) = 20s - 15t$ 2.c) $10(p + s - 3) = 10p + 10s - 30$ 3. Write each as a single fraction: 3.a) $\frac{3x}{4} + \frac{5x}{6}$ Find common denominator $12$: $$\frac{3x}{4} = \frac{3x \times 3}{4 \times 3} = \frac{9x}{12}, \quad \frac{5x}{6} = \frac{5x \times 2}{6 \times 2} = \frac{10x}{12}$$ Add: $$\frac{9x}{12} + \frac{10x}{12} = \frac{19x}{12}$$ 3.b) $2f + \frac{2}{3}f$ Rewrite $2f$ as $\frac{6}{3}f$: $$\frac{6}{3}f + \frac{2}{3}f = \frac{6f + 2f}{3} = \frac{8f}{3}$$ 4. Form and simplify expressions for perimeter: 4.a) Rectangle with sides $n$ and $n + 2$ Perimeter formula: $P = 2(\text{length} + \text{width})$ $$P = 2(n + (n + 2)) = 2(2n + 2) = 4n + 4$$ 4.b) Pentagon with sides $y, y, x, x, 2x$ Sum sides: $$y + y + x + x + 2x = 2y + 2x + 2x = 2y + 4x$$ 5. True or False: 5.a) $5c$ means $5 \times c$ True 5.b) $t + t + t = 3 + t$ False (should be $3t$) 5.c) $f + t + 2f = 3f + t$ True 6. Simplify expressions: 6.a) $4x + 2x = 6x$ 6.b) $4ab + c - 3ab + ab - 2c + 45$ Group like terms: $$(4ab - 3ab + ab) + (c - 2c) + 45 = 2ab - c + 45$$ 6.c) $8 - 3n + 5 + 6n$ Combine constants and like terms: $$(8 + 5) + (-3n + 6n) = 13 + 3n$$ 6.d) $5a - 6b + 8c - 2b - 3c - 12a$ Group like terms: $$(5a - 12a) + (-6b - 2b) + (8c - 3c) = -7a - 8b + 5c$$ 6.f) $t^2 + 2t + 3 - 5t^2 + 19$ Group like terms: $$(t^2 - 5t^2) + 2t + (3 + 19) = -4t^2 + 2t + 22$$ 6.g) $4g + 16 - 9 - 6g$ Combine like terms: $$(4g - 6g) + (16 - 9) = -2g + 7$$ 6.h) $6a^2 - 5a + a^2 + 3a + 4$ Group like terms: $$(6a^2 + a^2) + (-5a + 3a) + 4 = 7a^2 - 2a + 4$$ 6.i) $10 + 5u - 3 - 9u + 7$ Combine constants and like terms: $$(10 - 3 + 7) + (5u - 9u) = 14 - 4u$$ 7. Write as a single fraction: 7.a) $\frac{2m}{3} + \frac{5m}{3} = \frac{7m}{3}$ 7.b) $4d + \frac{2}{3}d$ Rewrite $4d$ as $\frac{12}{3}d$: $$\frac{12}{3}d + \frac{2}{3}d = \frac{14d}{3}$$ 8. Expand: 8.a) $4(6r - 7s + t) = 24r - 28s + 4t$ 8.b) $6(4m - 3n) = 24m - 18n$ 8.c) $5(p + 2r - 3) = 5p + 10r - 15$ 9. Expression for area of rectangle with width $7a + 4$ and height $10$: Area formula: $A = \text{width} \times \text{height}$ With brackets: $$A = (7a + 4) \times 10$$ Without brackets (expanded): $$A = 10 \times 7a + 10 \times 4 = 70a + 40$$ 10. Simplify: 10.a) $4f + 6g + 3f + 1 + 3g + 2$ Group like terms: $$(4f + 3f) + (6g + 3g) + (1 + 2) = 7f + 9g + 3$$ 10.b) $3m + 2n + 5m + 6 + n + 2m + 3$ Group like terms: $$(3m + 5m + 2m) + (2n + n) + (6 + 3) = 10m + 3n + 9$$ 10.c) $t^2 + 2t + 3 - 5t^2 + 19$ Group like terms: $$(t^2 - 5t^2) + 2t + (3 + 19) = -4t^2 + 2t + 22$$ 10.d) $4ab + c - 3ab + ab - 2c + 45$ Group like terms: $$(4ab - 3ab + ab) + (c - 2c) + 45 = 2ab - c + 45$