1. **Stating the problem:** Given points $A=(0,0)$, $B=(4,0)$, and $C=(0,4)$, find the equation of the line passing through points $A$ and $B$. 2. **Formula used:** The equation of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the slope-intercept form: $$y = mx + b$$ where the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ and $b$ is the y-intercept. 3. **Calculate the slope $m$:** $$m = \frac{0 - 0}{4 - 0} = \frac{0}{4} = 0$$ 4. **Find the y-intercept $b$:** Since point $A$ is at $(0,0)$, the y-intercept is $b=0$. 5. **Write the equation:** $$y = 0 \cdot x + 0 = 0$$ 6. **Interpretation:** The line through points $A$ and $B$ is the x-axis, where $y=0$ for all $x$. 1. **Stating the problem:** Find the equation of the line passing through points $A$ and $C$. 2. **Calculate the slope $m$:** $$m = \frac{4 - 0}{0 - 0} = \frac{4}{0}$$ Division by zero indicates a vertical line. 3. **Equation of vertical line:** The line passes through $x=0$. 4. **Write the equation:** $$x = 0$$ 1. **Stating the problem:** Find the equation of the line passing through points $B$ and $C$. 2. **Calculate the slope $m$:** $$m = \frac{4 - 0}{0 - 4} = \frac{4}{-4} = -1$$ 3. **Use point-slope form:** $$y - y_1 = m(x - x_1)$$ Using point $B=(4,0)$: $$y - 0 = -1(x - 4)$$ 4. **Simplify:** $$y = -x + 4$$ **Final answers:** - Line $AB$: $$y = 0$$ - Line $AC$: $$x = 0$$ - Line $BC$: $$y = -x + 4$$