1. **Problem statement:** Show that the Cartesian plane is a model of incidence geometry, i.e., it satisfies the axioms of incidence geometry. 2. **Axiom 1: Any two distinct points lie on exactly one line.** - In the Cartesian plane, points are pairs $(x,y)$. - Given two distinct points $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ with $x_1 \neq x_2$ or $y_1 \neq y_2$, there is exactly one line passing through both. - The line can be expressed uniquely by the slope-intercept form or standard form: $$ y = m x + b $$ where the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ (if $x_1 \neq x_2$) and intercept $b$ is computed accordingly. - If $x_1 = x_2$, the line is vertical $x = x_1$. 3. **Axiom 2: Every line has at least two distinct points.** - In the Cartesian plane, each line includes infinitely many points. - For example, the horizontal line $y = c$ includes points $(x, c)$ for all real $x$. - So each line has infinitely many points, definitely at least two. 4. **Axiom 3: There exist at least three non-collinear points.** - Choose points $P_1 = (0,0)$, $P_2 = (1,0)$, and $P_3 = (0,1)$. - These three points do not all lie on the same line, so they are non-collinear. 5. **Conclusion:** The Cartesian plane with points as ordered pairs and lines as linear equations satisfies the axioms of incidence geometry. **Note on visual diagram:** - While a detailed diagram cannot be directly shown here, visualize the Cartesian plane with points plotted and the lines constructed between them as described. **Summary of axioms verified:** - Unique line through any two points. - Each line contains multiple points. - Existence of non-collinear points. This shows the Cartesian plane is a model of incidence geometry.