1. **Stating the problem:** Find the equation of a line using different methods: (a) Two points (b) Slope and a point (c) Slope and y-intercept (d) X- and y-intercepts 2. **Formula and rules:** - Slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ - Point-slope form of a line: $$y - y_1 = m(x - x_1)$$ - Slope-intercept form: $$y = mx + b$$ where $b$ is the y-intercept. - To find the equation from x- and y-intercepts $(a,0)$ and $(0,b)$: $$\text{slope } m = \frac{0 - b}{a - 0} = -\frac{b}{a}$$ Equation: $$y = m x + b$$ 3. **(a) Using two points:** Choose points $(5,770)$ and $(15,2270)$ from the problem. Calculate slope: $$m = \frac{2270 - 770}{15 - 5} = \frac{1500}{10} = 150$$ Use point-slope form with point $(5,770)$: $$y - 770 = 150(x - 5)$$ Simplify: $$y - 770 = 150x - 750$$ $$y = 150x + 20$$ 4. **(b) Using slope and a point:** Given slope $m=150$ and point $(15,2270)$: $$y - 2270 = 150(x - 15)$$ Simplify: $$y - 2270 = 150x - 2250$$ $$y = 150x + 20$$ 5. **(c) Using slope and y-intercept:** Given slope $m=150$ and y-intercept $b=20$: Equation is directly: $$y = 150x + 20$$ 6. **(d) Using x- and y-intercepts:** From the equation $y = 150x + 20$, find x-intercept by setting $y=0$: $$0 = 150x + 20$$ $$150x = -20$$ $$x = \frac{-20}{150} = -\frac{2}{15}$$ So x-intercept is $\left(-\frac{2}{15}, 0\right)$ and y-intercept is $(0,20)$. Calculate slope: $$m = \frac{0 - 20}{-\frac{2}{15} - 0} = \frac{-20}{-\frac{2}{15}} = -20 \times -\frac{15}{2} = 150$$ Equation: $$y = 150x + 20$$ **Final answer:** The equation of the line is $$y = 150x + 20$$ This equation is consistent across all methods.