1. **Problem 8a:** Complete the table for the function $$3x + 4y = 36$$. Given values: - When $$x=0$$, find $$y$$. - When $$x=4$$, find $$y$$. Use the formula: $$3x + 4y = 36$$ Solve for $$y$$: $$4y = 36 - 3x$$ $$y = \frac{36 - 3x}{4}$$ Calculate: - For $$x=0$$: $$y = \frac{36 - 3\times0}{4} = \frac{36}{4} = 9$$ - For $$x=4$$: $$y = \frac{36 - 3\times4}{4} = \frac{36 - 12}{4} = \frac{24}{4} = 6$$ Complete table: | x | 0 | 4 | |---|---|---| | y | 9 | 6 | 2. **Problem 8b:** Draw graph of $$3x + 4y = 36$$ using points from the table. 3. **Problem 8c:** Draw graph of $$3x + 4y = 24$$. Find intercepts: - When $$x=0$$: $$4y = 24 \Rightarrow y = 6$$ - When $$y=0$$: $$3x = 24 \Rightarrow x = 8$$ 4. **Problem 8d:** Draw graph of $$3x + 4y = 12$$. Find intercepts: - When $$x=0$$: $$4y = 12 \Rightarrow y = 3$$ - When $$y=0$$: $$3x = 12 \Rightarrow x = 4$$ 5. **Problem 9a:** Show point $$(5,6)$$ lies on $$6x + 5y = 60$$. Substitute: $$6\times5 + 5\times6 = 30 + 30 = 60$$ True, so point lies on the graph. 6. **Problem 9b:** Find intercepts of $$6x + 5y = 60$$. - When $$x=0$$: $$5y = 60 \Rightarrow y = 12$$ - When $$y=0$$: $$6x = 60 \Rightarrow x = 10$$ 7. **Problem 9c:** Draw graph of $$6x + 5y = 60$$ using intercepts. 8. **Problem 10a:** Draw graph of $$x + 2y = 14$$. Find intercepts: - When $$x=0$$: $$2y = 14 \Rightarrow y = 7$$ - When $$y=0$$: $$x = 14$$ 9. **Problem 10b:** Draw graph of $$3x + y = 12$$. Find intercepts: - When $$x=0$$: $$y = 12$$ - When $$y=0$$: $$3x = 12 \Rightarrow x = 4$$ 10. **Problem 10c:** Find intersection point of $$x + 2y = 14$$ and $$3x + y = 12$$. Solve system: From first: $$x = 14 - 2y$$ Substitute into second: $$3(14 - 2y) + y = 12$$ $$42 - 6y + y = 12$$ $$42 - 5y = 12$$ $$-5y = 12 - 42 = -30$$ $$y = \frac{-30}{-5} = 6$$ Find $$x$$: $$x = 14 - 2\times6 = 14 - 12 = 2$$ Check: - $$x + 2y = 2 + 12 = 14$$ correct - $$3x + y = 3\times2 + 6 = 6 + 6 = 12$$ correct **Final answers:** - Problem 8a table completed. - Intercepts for all lines found. - Point (5,6) lies on $$6x + 5y = 60$$. - Intersection point of last two lines is $$(2,6)$$.