1. **Stating the problem:** We are given three pairs of linear equations and asked to analyze their relationships and solve the first pair. 2. **Analyzing pair (a):** Equations: $$2x - 3y + 5 = 0$$ and $$y = \frac{2}{3}x + \frac{5}{6}$$ Rewrite the first equation in slope-intercept form: $$2x - 3y + 5 = 0 \implies -3y = -2x - 5 \implies y = \frac{2}{3}x + \frac{5}{3}$$ Notice the second equation has intercept $\frac{5}{6}$, which is different from $\frac{5}{3}$, so they are not the same line. However, the user states they represent the same line, so let's check carefully: Multiply the second equation by 6 to clear denominators: $$6y = 4x + 5$$ Multiply the first equation by 2: $$4x - 6y + 10 = 0 \implies 4x - 6y = -10$$ Rewrite: $$4x - 6y = -10$$ Compare with: $$6y = 4x + 5 \implies 4x - 6y = -5$$ They differ by a constant term, so they are not the same line but parallel lines with different intercepts. 3. **Solving pair (a):** Since the user states they represent the same line, let's verify if the second equation is equivalent to the first. Rewrite second equation: $$y = \frac{2}{3}x + \frac{5}{6}$$ Multiply both sides by 3: $$3y = 2x + \frac{5}{2}$$ Multiply both sides by 2: $$6y = 4x + 5$$ Rewrite first equation multiplied by 2: $$4x - 6y + 10 = 0 \implies 4x - 6y = -10$$ Since $4x - 6y = -10$ and $4x - 6y = -5$ are not equal, the lines are not the same. Therefore, the user’s hint is incorrect; the lines are parallel but distinct. 4. **Summary for (a):** The lines are parallel and do not intersect. 5. **Final answer for (a):** The lines are parallel and distinct; no solution exists for the system. **Note:** Since the user asked multiple parts but per instructions only the first problem is solved, the rest are ignored.