1. Problem: Solve the equations by distributing first, then solving for the variable. (a) Solve $x = 3(5 - x) + 1$ 2. Use the distributive property: $a(b + c) = ab + ac$ 3. Distribute $3$ over $(5 - x)$: $$x = 3 \times 5 - 3 \times x + 1 = 15 - 3x + 1$$ 4. Simplify the right side: $$x = 16 - 3x$$ 5. Add $3x$ to both sides to get all $x$ terms on one side: $$x + 3x = 16 - \cancel{3x} + \cancel{3x}$$ $$4x = 16$$ 6. Divide both sides by $4$: $$\frac{4x}{\cancel{4}} = \frac{16}{\cancel{4}}$$ $$x = 4$$ (a) Answer: $x = 4$ 1. Problem: Solve $4(x + 3) = 2(x + 6) - x$ 2. Distribute on both sides: $$4x + 12 = 2x + 12 - x$$ 3. Simplify the right side: $$4x + 12 = x + 12$$ 4. Subtract $12$ from both sides: $$4x + 12 - 12 = x + 12 - 12$$ $$4x = x$$ 5. Subtract $x$ from both sides: $$4x - x = x - x$$ $$3x = 0$$ 6. Divide both sides by $3$: $$\frac{3x}{\cancel{3}} = \frac{0}{\cancel{3}}$$ $$x = 0$$ (a,c) Answer: $x = 0$ 1. Problem: Rearrange and isolate $y$ in $x + 2y + 5 = 0$ 2. Subtract $x$ and $5$ from both sides: $$x + 2y + 5 - x - 5 = 0 - x - 5$$ $$2y = -x - 5$$ 3. Divide both sides by $2$: $$\frac{2y}{\cancel{2}} = \frac{-x - 5}{\cancel{2}}$$ $$y = -\frac{x}{2} - \frac{5}{2}$$ (a,iii) Answer: $y = -\frac{x}{2} - \frac{5}{2}$