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

**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.