1. **State the problem:** We are given the system of linear equations: $$2y = x + 10$$ $$3y = 3x + 15$$ We need to determine which statements about this system are true. 2. **Rewrite each equation in slope-intercept form $y = mx + b$:** For the first equation: $$2y = x + 10$$ Divide both sides by 2: $$y = \frac{\cancel{2}y}{\cancel{2}} = \frac{x + 10}{2} = \frac{x}{2} + 5$$ For the second equation: $$3y = 3x + 15$$ Divide both sides by 3: $$y = \frac{\cancel{3}y}{\cancel{3}} = \frac{3x + 15}{3} = x + 5$$ 3. **Identify slopes and y-intercepts:** - First line: slope $m_1 = \frac{1}{2}$, y-intercept $b_1 = 5$ - Second line: slope $m_2 = 1$, y-intercept $b_2 = 5$ 4. **Analyze the statements:** - "The system has one solution." Since slopes are different ($\frac{1}{2} \neq 1$), lines intersect at exactly one point. **True**. - "The system graphs parallel lines." Parallel lines have equal slopes. Here, slopes differ. **False**. - "Both lines have the same slope." Slopes are $\frac{1}{2}$ and $1$. **False**. - "Both lines have the same y-intercept." Both have $b=5$. **True**. - "The equations graph the same line." For same line, slopes and intercepts must be equal. Slopes differ. **False**. - "The solution is the intersection of the 2 lines." By definition, solution to system is intersection point. **True**. 5. **Final answers:** - The system has one solution. ✔ - Both lines have the same y-intercept. ✔ - The solution is the intersection of the 2 lines. ✔ "desmos" field is minimal as no graph requested.