1. Calculate angles T, W, X, Y, and Z given two intersecting parallel lines with angles 55° and 65°. Step 1: Recognize that alternate interior angles and corresponding angles are equal when lines are parallel. Step 2: Angle at T is vertically opposite to 55°, so $T = 55^\circ$. Step 3: Angle at W is adjacent to 65°, so supplementary angle to 65° is $180^\circ - 65^\circ = 115^\circ$, hence $W = 115^\circ$. Step 4: Angle at X complements to T, so $X = 180^\circ - 55^\circ = 125^\circ$. Step 5: Angle Y equals to corresponding angle 65°, so $Y = 65^\circ$. Step 6: Angle Z is vertically opposite to W, so $Z = 115^\circ$. 2. For the regression equation $S = 2.2 A + 1400$: a) The rate sales increase per K1 spent on advertising is the coefficient of A, i.e., $2.2$. b) Predicted sales if K10,000 is spent: $$ S = 2.2 \times 10000 + 1400 = 22000 + 1400 = 23400 $$ c) To find advertising spend for sales of K16,000: $$ 16000 = 2.2 A + 1400 \Rightarrow 2.2 A = 16000 - 1400 = 14600 \Rightarrow A = \frac{14600}{2.2} = 6636.36 $$ d) Profit from K30,000 advertising spend is not given explicitly, assuming profit equals sales minus advertising: Calculate sales: $$ S = 2.2 \times 30000 + 1400 = 66000 + 1400 = 67400 $$ Profit: $$ 67400 - 30000 = 37400 $$ 3. Solve system: $a)$ Solve simultaneously: $$ 2x - y = 5 \quad (1) $$ $$ x + 4y = 7 \quad (2) $$ From (1): $y = 2x - 5$ Substitute into (2): $$ x + 4(2x - 5) = 7 $$ $$ x + 8x - 20 = 7 $$ $$ 9x = 27 \Rightarrow x = 3 $$ Then, $$ y = 2(3) - 5 = 6 - 5 = 1 $$ Solution: $(x, y) = (3, 1)$ $b)$ Solve quadratic equation using formula: $$ x^2 - 2x - 15 = 0 $$ Quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here $a=1, b=-2, c=-15$: $$ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)} = \frac{2 \pm \sqrt{4 + 60}}{2} = \frac{2 \pm \sqrt{64}}{2} $$ $$ = \frac{2 \pm 8}{2} $$ Solutions: $$ x = \frac{2 + 8}{2} = 5, \quad x = \frac{2 - 8}{2} = -3 $$ 4. Graph questions: a) Slope of straight line: Using points approximately (-10, ?) and (0, -3), slope = $\frac{-3 - y_1}{0 - (-10)}$, from graph y intercept is -3, estimate slope near 1 (positive slope from graph behavior). To find exactly: Using points (-4, 0) and (4, 2): $$ m = \frac{2 - 0}{4 - (-4)} = \frac{2}{8} = 0.25 $$ b) Parabola cuts y-axis at $(0, -2)$. c) Equation of straight line: Use point-slope form with slope $0.25$ and point (0, -3): $$ y = 0.25 x - 3 $$ d) Equation of parabola: Vertex is near (-2, -12), and passes (0, -2). Use vertex form: $$ y = a(x - h)^2 + k $$ $$ y = a(x + 2)^2 - 12 $$ At (0, -2): $$ -2 = a(0 + 2)^2 - 12 = 4a - 12 \Rightarrow 4a = 10 \Rightarrow a = 2.5 $$ So, $$ y = 2.5 (x + 2)^2 - 12 $$ e) X-intercepts of parabola: Set $y=0$: $$ 0 = 2.5 (x + 2)^2 - 12 $$ $$ 2.5 (x + 2)^2 = 12 $$ $$ (x + 2)^2 = \frac{12}{2.5} = 4.8 $$ $$ x + 2 = \pm \sqrt{4.8} = \pm 2.19 $$ $$ x = -2 \pm 2.19 $$ So, approximately: $$ x = 0.19 \text{ or } -4.19 $$