1. The problem is to list and explain the 7 cases of linear equations. 2. A linear equation in one variable is generally of the form $ax + b = 0$, where $a$ and $b$ are constants and $x$ is the variable. 3. The 7 cases of linear equations typically refer to different scenarios based on the values of $a$ and $b$ and the number of variables and equations involved. 4. Case 1: One variable, one equation, $a \neq 0$. - Equation: $ax + b = 0$ - Solution: $x = -\frac{b}{a}$ 5. Case 2: One variable, one equation, $a = 0$, $b \neq 0$. - Equation: $0 \cdot x + b = 0$ or $b = 0$ - No solution because $b \neq 0$. 6. Case 3: One variable, one equation, $a = 0$, $b = 0$. - Equation: $0 \cdot x + 0 = 0$ - Infinite solutions because any $x$ satisfies the equation. 7. Case 4: Two variables, one equation. - Equation: $ax + by = c$ - Infinite solutions forming a line in the plane. 8. Case 5: Two variables, two equations, consistent and independent. - System: $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$ - Unique solution if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. 9. Case 6: Two variables, two equations, consistent and dependent. - System where one equation is a multiple of the other. - Infinite solutions. 10. Case 7: Two variables, two equations, inconsistent. - System with parallel lines, no solution. These cases cover the typical scenarios encountered in linear equations and systems.