1. Let's start by understanding what an "adequate example" means in math: it is a clear, simple instance that illustrates a concept or problem. 2. Example 1: Solve the linear equation $2x + 3 = 7$. 3. Use the formula for solving linear equations: isolate $x$ by subtracting 3 from both sides and then dividing by 2. 4. Step-by-step: $$2x + 3 = 7$$ Subtract 3: $$2x + \cancel{3} - \cancel{3} = 7 - 3$$ $$2x = 4$$ Divide both sides by 2: $$\frac{2x}{\cancel{2}} = \frac{4}{\cancel{2}}$$ $$x = 2$$ 5. Example 2: Factor the quadratic expression $x^2 - 5x + 6$. 6. Use factoring rules: find two numbers that multiply to 6 and add to -5. 7. The numbers are -2 and -3, so: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ 8. Example 3: Calculate the slope of the line passing through points $(1,2)$ and $(3,6)$. 9. Use the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ $$m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$$ These examples cover solving equations, factoring, and slope calculation, which are fundamental algebra concepts.