1. **State the problem:** Lin has $10 to spend on brown rice and beans. Brown rice costs 2 per pound, beans cost 1.60 per pound. Let $b$ be pounds of beans and $r$ be pounds of rice. The total cost equation is: $$2r + 1.60b = 10$$ 2. **Write the equation relating the two variables:** The equation is already given: $$2r + 1.60b = 10$$ 3. **Rearrange the equation so $b$ is the independent variable:** Isolate $b$: $$1.60b = 10 - 2r$$ Divide both sides by 1.60: $$b = \frac{10 - 2r}{1.60}$$ Intermediate step showing cancellation: $$b = \frac{\cancel{10} - 2r}{\cancel{1.60}}$$ 4. **Rearrange the equation so $r$ is the independent variable:** Isolate $r$: $$2r = 10 - 1.60b$$ Divide both sides by 2: $$r = \frac{10 - 1.60b}{2}$$ Intermediate step showing cancellation: $$r = \frac{\cancel{10} - 1.60b}{\cancel{2}}$$ 5. **Solve each equation and check your answer:** **Equation 1:** $$2x + 4(3 - 2x) = \frac{3(2x+2)}{6} + 4$$ Step 1: Expand and simplify left side: $$2x + 12 - 8x = \frac{3(2x+2)}{6} + 4$$ $$-6x + 12 = \frac{6x + 6}{6} + 4$$ Step 2: Simplify right side: $$-6x + 12 = x + 1 + 4$$ $$-6x + 12 = x + 5$$ Step 3: Bring variables to one side: $$-6x - x = 5 - 12$$ $$-7x = -7$$ Step 4: Divide both sides by -7: $$x = \frac{-7}{-7} = 1$$ Check: Left: $2(1) + 4(3 - 2(1)) = 2 + 4(1) = 2 + 4 = 6$ Right: $\frac{3(2(1)+2)}{6} + 4 = \frac{3(4)}{6} + 4 = 2 + 4 = 6$ Correct. **Equation 2:** $$4z + 5 = -3z - 8$$ Step 1: Bring variables to one side: $$4z + 3z = -8 - 5$$ $$7z = -13$$ Step 2: Divide both sides by 7: $$z = \frac{-13}{7}$$ Check: Left: $4(-13/7) + 5 = -52/7 + 35/7 = -17/7$ Right: $-3(-13/7) - 8 = 39/7 - 56/7 = -17/7$ Correct. **Equation 3:** $$\frac{1}{2} - \frac{1}{8}q = \frac{q - 1}{4}$$ Step 1: Multiply all terms by 8 to clear denominators: $$8 \times \left(\frac{1}{2} - \frac{1}{8}q\right) = 8 \times \frac{q - 1}{4}$$ $$4 - q = 2(q - 1)$$ Step 2: Expand right side: $$4 - q = 2q - 2$$ Step 3: Bring variables to one side: $$4 + 2 = 2q + q$$ $$6 = 3q$$ Step 4: Divide both sides by 3: $$q = 2$$ Check: Left: $\frac{1}{2} - \frac{1}{8} \times 2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$ Right: $\frac{2 - 1}{4} = \frac{1}{4}$ Correct. **Final answers:** $$b = \frac{10 - 2r}{1.60}$$ $$r = \frac{10 - 1.60b}{2}$$ $$x = 1$$ $$z = -\frac{13}{7}$$ $$q = 2$$