Ine Qualities 2.a) Draw the groph of $$ 4 x+y=8 $$ $$ x+4 y=8 $$ and $$ x+y=2 $$

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**Step 1. Rewrite each equation in slope-intercept form.**

1. For $$4x + y = 8$$:
   - Solve for $$y$$:
     $$
     y = 8 - 4x
     $$
   - **Intercepts:**
     - When $$x = 0$$: $$y = 8$$ $$\rightarrow (0,8)$$.
     - When $$y = 0$$: $$0 = 8 - 4x \Rightarrow 4x = 8 \Rightarrow x = 2$$ $$\rightarrow (2,0)$$.

2. For $$x + 4y = 8$$:
   - Solve for $$y$$:
     $$
     4y = 8 - x \quad \Rightarrow \quad y = 2 - \frac{x}{4}
     $$
   - **Intercepts:**
     - When $$x = 0$$: $$y = 2$$ $$\rightarrow (0,2)$$.
     - When $$y = 0$$: $$0 = 2 - \frac{x}{4} \Rightarrow \frac{x}{4} = 2 \Rightarrow x = 8$$ $$\rightarrow (8,0)$$.

3. For $$x + y = 2$$:
   - Solve for $$y$$:
     $$
     y = 2 - x
     $$
   - **Intercepts:**
     - When $$x = 0$$: $$y = 2$$ $$\rightarrow (0,2)$$.
     - When $$y = 0$$: $$0 = 2 - x \Rightarrow x = 2$$ $$\rightarrow (2,0)$$.

---

**Step 2. Plot the Points and Draw the Lines**

1. **Graph of $$4x + y = 8$$:**
   - Plot the intercepts $$(0,8)$$ and $$(2,0)$$.
   - Draw a straight line through these points.
   - The line has a slope $$-4$$.

2. **Graph of $$x + 4y = 8$$:**
   - Plot the intercepts $$(0,2)$$ and $$(8,0)$$.
   - Draw a straight line through these points.
   - The line has a slope of $$-\frac{1}{4}$$.

3. **Graph of $$x + y = 2$$:**
   - Plot the intercepts $$(0,2)$$ and $$(2,0)$$.
   - Draw a straight line through these points.
   - The line has a slope of $$-1$$.

---

**Step 3. Verify the Intersection Points**

1. **Intersection of $$x + y = 2$$ and $$x + 4y = 8$$:**

- From $$x+y=2$$, express $$x$$ as:
     $$
     x = 2 - y
     $$
   - Substitute into $$x + 4y = 8$$:
     $$
     (2 - y) + 4y = 8 \quad \Rightarrow \quad 2 + 3y = 8 \quad \Rightarrow \quad 3y = 6 \quad \Rightarrow \quad y = 2
     $$
   - Substitute $$y = 2$$ into $$x = 2 - y$$:
     $$
     x = 2 - 2 = 0
     $$
   - Intersection point is $$(0,2)$$.

2. **Intersection of $$x + y = 2$$ and $$4x + y = 8$$:**

- From $$x+y=2$$, express $$y$$ as:
     $$
     y = 2 - x
     $$
   - Substitute into $$4x + y = 8$$:
     $$
     4x + (2 - x) = 8 \quad \Rightarrow \quad 3x + 2 = 8 \quad \Rightarrow \quad 3x = 6 \quad \Rightarrow \quad x = 2
     $$
   - Then, $$y = 2 - 2 = 0$$.
   - Intersection point is $$(2,0)$$.

3. **Intersection of $$4x + y = 8$$ and $$x + 4y = 8$$:**

- We already have two intersections from above with the third line, but we find this intersection as well.
   - Solve the system:
     $$
     \begin{cases}
     4x + y = 8 \
     x + 4y = 8
     \end{cases}
     $$
   - Multiply the second equation by 4:
     $$
     4x + 16y = 32
     $$
   - Subtract the first equation $$ (4x+y=8) $$ from this:
     $$
     (4x + 16y) - (4x + y) = 32 - 8 \quad \Rightarrow \quad 15y = 24 \quad \Rightarrow \quad y = \frac{24}{15} = \frac{8}{5}
     $$
   - Substitute $$ y = \frac{8}{5} $$ into $$4x + y = 8$$:
     $$
     4x + \frac{8}{5} = 8 \quad \Rightarrow \quad 4x = 8 - \frac{8}{5} = \frac{40}{5} - \frac{8}{5} = \frac{32}{5} \quad \Rightarrow \quad x = \frac{32}{20} = \frac{8}{5}
     $$
   - Intersection point is $$\left(\frac{8}{5}, \frac{8}{5}\right)$$.

---

**Step 4. Summary of Key Points**

- **Line $$4x+y=8$$:** Intercepts at $$(0,8)$$ and $$(2,0)$$; slope $$-4$$.
- **Line $$x+4y=8$$:** Intercepts at $$(0,2)$$ and $$(8,0)$$; slope $$-\frac{1}{4}$$.
- **Line $$x+y=2$$:** Intercepts at $$(0,2)$$ and $$(2,0)$$; slope $$-1$$.

- **Intersection Points:**
  - $$x+y=2$$ and $$x+4y=8$$ intersect at $$(0,2)$$.
  - $$x+y=2$$ and $$4x+y=8$$ intersect at $$(2,0)$$.
  - $$4x+y=8$$ and $$x+4y=8$$ intersect at $$\left(\frac{8}{5}, \frac{8}{5}\right)$$.

---

**Step 5. Plotting on the Coordinate Plane**

- Draw a coordinate system with appropriate scaling.
- Plot each intercept:
  - For $$4x+y=8$$: mark $$(0,8)$$ and $$(2,0)$$.
  - For $$x+4y=8$$: mark $$(0,2)$$ and $$(8,0)$$.
  - For $$x+y=2$$: mark $$(0,2)$$ and $$(2,0)$$.  
- Draw the three straight lines through their respective intercepts.
- Mark the intersection points clearly on the graph.

These steps allow you to accurately graph the three lines and identify their intersections.
To graph the equations $$4x + y = 8$$, $$x + 4y = 8$$, and $$x + y = 2$$, follow these steps: 1. **Convert each equation to slope-intercept form:** - $$4x + y = 8$$ becomes $$y = 8 - 4x$$. - $$x + 4y = 8$$ becomes $$y = 2 - \frac{x}{4}$$. - $$x + y = 2$$ becomes $$y = 2 - x$$. 2. **Find the intercepts for each line:** - For $$4x + y = 8$$: intercepts at $$(0,8)$$ and $$(2,0)$$. - For $$x + 4y = 8$$: intercepts at $$(0,2)$$ and $$(8,0)$$. - For $$x + y = 2$$: intercepts at $$(0,2)$$ and $$(2,0)$$. 3. **Plot the intercepts on a coordinate plane and draw each line through them.** 4. **Identify the intersection points:** - $$x + y = 2$$ and $$x + 4y = 8$$ intersect at $$(0,2)$$. - $$x + y = 2$$ and $$4x + y = 8$$ intersect at $$(2,0)$$. - $$4x + y = 8$$ and $$x + 4y = 8$$ intersect at $$\left(\frac{8}{5}, \frac{8}{5}\right)$$. By following these steps, you can accurately graph the three equations and locate their intersection points.

Ine Qualities 2.a) Draw the groph of \( 4 x+y=8 \) \( x+4 y=8 \) and \( x+y=2 \)

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