1. The problem asks to determine the number of solutions (0, 1, or infinite) for each given equation after substitution or elimination in a system. 2. Important rules: - If an equation simplifies to a true statement involving variables (like $x = a$), there is exactly 1 solution. - If an equation simplifies to a false statement (like $8 = 0$), there are 0 solutions. - If an equation simplifies to a true statement without variables (like $8 = 8$), there are infinite solutions. 3. Analyze each equation: a. $5x = 12$ - Solve for $x$: $x = \frac{12}{5}$ - This is a single value, so 1 solution. b. $\frac{2}{x} = 6$ - Solve for $x$: multiply both sides by $x$ (assuming $x \neq 0$): $$2 = 6x$$ - Then $x = \frac{2}{6} = \frac{1}{3}$ - One solution. c. $\frac{2}{x} = 0$ - For a fraction to be zero, numerator must be zero, but numerator is 2 (nonzero). - No solution. d. $\frac{x}{2} = 0$ - Multiply both sides by 2: $$\cancel{2} \times \frac{x}{\cancel{2}} = 0 \times 2$$ $$x = 0$$ - One solution. e. $8 = 0$ - False statement, no solution. f. $8 = 8$ - True statement, infinite solutions. Final answers: a. 1 solution b. 1 solution c. 0 solutions d. 1 solution e. 0 solutions f. infinite solutions