1. **State the problem:** We are given two pairs of lines and need to determine if each pair is parallel, intersecting, or the same line, and identify the number of solutions for each system. 2. **Recall the rule:** Two lines are parallel if they have the same slope but different intercepts. They are the same line if they have the same slope and intercept. They intersect if their slopes are different. 3. **First pair:** $$y = x + 6$$ $$y = 40 - 6x$$ Rewrite second line as $$y = -6x + 40$$ Slopes: first line $m_1 = 1$, second line $m_2 = -6$ Since $m_1 \neq m_2$, lines intersect. 4. **Number of solutions for first pair:** One unique solution (the point of intersection). 5. **Second pair:** $$y = -3x + 1$$ $$y = 3x + 1$$ Slopes: first line $m_1 = -3$, second line $m_2 = 3$ Since $m_1 \neq m_2$, lines intersect. 6. **Number of solutions for second pair:** One unique solution. **Final answers:** - Relationship for first pair: intersecting - Number of solutions for first pair: 1 - Relationship for second pair: intersecting - Number of solutions for second pair: 1