1. **Evaluate** $y = k(3 + m)^2$ when $k = -2$ and $m = 9$. Step 1: Substitute the values of $k$ and $m$ into the expression: $$y = -2(3 + 9)^2$$ Step 2: Simplify inside the parentheses: $$y = -2(12)^2$$ Step 3: Square 12: $$y = -2 \times 144$$ Step 4: Multiply: $$y = -288$$ **Answer:** $y = -288$ 2. **Identify if the relations are linear or not.** (a) $w = (5 + s)^2$ Step 1: Expand the square: $$w = (5 + s)^2 = 25 + 10s + s^2$$ Step 2: Notice the $s^2$ term, which is a squared variable. Step 3: Since linear relations cannot have variables with exponents other than 1, this is **not linear**. (b) $y = 3x - \frac{1}{2}$ Step 1: The expression is in the form $y = mx + b$ where $m=3$ and $b = -\frac{1}{2}$. Step 2: This is a linear equation because the variable $x$ is to the first power and the equation forms a straight line. **Answer:** (a) Not linear, (b) Linear 3. **Rectangular swimming pool problem:** Given: Length $= 2 \times$ width, width $= 5x$, perimeter $= 180$ meters. (a) Create an equation for $x$. Step 1: Express length in terms of $x$: $$\text{Length} = 2 \times 5x = 10x$$ Step 2: Recall perimeter formula for rectangle: $$P = 2(\text{Length} + \text{Width})$$ Step 3: Substitute values: $$180 = 2(10x + 5x)$$ Step 4: Simplify inside parentheses: $$180 = 2(15x)$$ Step 5: Multiply: $$180 = 30x$$ This is the equation to solve. (b) Solve for $x$. Step 1: Divide both sides by 30: $$\frac{180}{\cancel{30}} = \frac{30x}{\cancel{30}}$$ Step 2: Simplify: $$6 = x$$ (c) Find width and length. Step 1: Width: $$5x = 5 \times 6 = 30 \text{ meters}$$ Step 2: Length: $$10x = 10 \times 6 = 60 \text{ meters}$$ **Answer:** Width = 30 meters, Length = 60 meters