1. **State the problem:** Solve the first system of linear equations: $$3x - y = 7$$ $$2x + 5y + 1 = 0$$ 2. **Rewrite the second equation:** $$2x + 5y = -1$$ 3. **Express $y$ from the first equation:** $$3x - y = 7 \implies y = 3x - 7$$ 4. **Substitute $y$ into the second equation:** $$2x + 5(3x - 7) = -1$$ $$2x + 15x - 35 = -1$$ $$17x - 35 = -1$$ 5. **Solve for $x$:** $$17x = 34 \implies x = 2$$ 6. **Find $y$ using $x=2$:** $$y = 3(2) - 7 = 6 - 7 = -1$$ 7. **Final solution for the first system:** $$(x, y) = (2, -1)$$ --- 1. **State the problem:** Solve the second system of linear equations: $$x + 3y = 12$$ $$2x - 3y = 12$$ 2. **Add the two equations to eliminate $y$:** $$(x + 3y) + (2x - 3y) = 12 + 12$$ $$3x = 24 \implies x = 8$$ 3. **Substitute $x=8$ into the first equation:** $$8 + 3y = 12 \implies 3y = 4 \implies y = \frac{4}{3}$$ 4. **Final solution for the second system:** $$(x, y) = \left(8, \frac{4}{3}\right)$$ --- 1. **State the problem:** Shade the region bounded by the line $$2x - 3y = 2$$ and the coordinate axes. 2. **Rewrite the line in terms of $y$:** $$2x - 3y = 2 \implies 3y = 2x - 2 \implies y = \frac{2x - 2}{3}$$ 3. **Find intercepts:** - When $x=0$, $$y = \frac{2(0) - 2}{3} = -\frac{2}{3}$$ (negative, so no positive intercept on y-axis) - When $y=0$, $$2x - 3(0) = 2 \implies x = 1$$ 4. **Since the y-intercept is negative, the bounded region with axes and line is a triangle with vertices at:** $$(0,0), (1,0), \text{and the point where the line crosses the y-axis if positive}$$ 5. **Check where the line crosses the y-axis positively:** Set $x=1$ in the line equation: $$y = \frac{2(1) - 2}{3} = 0$$ 6. **Check the region bounded by the line and axes:** The line crosses the x-axis at $(1,0)$ and the y-axis at $(0,-\frac{2}{3})$ (below x-axis), so the bounded region is the triangle formed by the x-axis, y-axis, and the line segment from $(0,0)$ to $(1,0)$. 7. **Shade the triangular region bounded by:** - The x-axis ($y=0$) - The y-axis ($x=0$) - The line $2x - 3y = 2$ --- **Summary:** - First system solution: $(2, -1)$ - Second system solution: $(8, \frac{4}{3})$ - Shaded region: triangle bounded by $x=0$, $y=0$, and $2x - 3y = 2$