1. **Problem Statement:** We need to find and indicate the solution region for the system of linear inequalities: $$3x + 7y \leq 21$$ $$x - y \geq 2$$ 2. **Understanding the inequalities:** - The first inequality, $3x + 7y \leq 21$, represents all points $(x,y)$ on or below the line $3x + 7y = 21$. - The second inequality, $x - y \geq 2$, represents all points $(x,y)$ on or above the line $x - y = 2$. 3. **Rewrite the boundary lines in slope-intercept form:** - For $3x + 7y = 21$: $$7y = 21 - 3x$$ $$y = \frac{21 - 3x}{7} = 3 - \frac{3}{7}x$$ - For $x - y = 2$: $$-y = 2 - x$$ $$y = x - 2$$ 4. **Determine the solution regions for each inequality:** - For $3x + 7y \leq 21$, the region is below or on the line $y = 3 - \frac{3}{7}x$. - For $x - y \geq 2$, the region is above or on the line $y = x - 2$. 5. **Find the intersection points of the boundary lines to understand the feasible region:** Set $$3 - \frac{3}{7}x = x - 2$$ Multiply both sides by 7: $$21 - 3x = 7x - 14$$ Bring all terms to one side: $$21 + 14 = 7x + 3x$$ $$35 = 10x$$ $$x = 3.5$$ Substitute back to find $y$: $$y = 3.5 - 2 = 1.5$$ 6. **Interpretation:** The solution region is the intersection of the half-planes below the first line and above the second line. This region lies between the two lines and includes the point $(3.5, 1.5)$ where they intersect. 7. **Summary:** - Shade the area below or on $y = 3 - \frac{3}{7}x$. - Shade the area above or on $y = x - 2$. - The solution region is where these shaded areas overlap, which is the region between the two lines including their boundaries. **Final answer:** The solution region is the set of points satisfying both $3x + 7y \leq 21$ and $x - y \geq 2$, i.e., the area between and including the lines $y = 3 - \frac{3}{7}x$ and $y = x - 2$.