1. **State the problem:** Solve the equation $2x - 6 = 4x + 2$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable on one side by performing the same operation on both sides. 3. **Step 1:** Move all $x$ terms to one side by subtracting $2x$ from both sides: $$2x - 6 - \cancel{2x} = 4x + 2 - \cancel{2x} \implies -6 = 2x + 2$$ 4. **Step 2:** Move constants to the other side by subtracting $2$ from both sides: $$-6 - 2 = 2x + 2 - 2 \implies -8 = 2x$$ 5. **Step 3:** Divide both sides by $2$ to solve for $x$: $$\frac{-8}{\cancel{2}} = \frac{2x}{\cancel{2}} \implies -4 = x$$ 6. **Final answer:** $x = -4$. --- 1. **State the problem:** Solve the equation $5x - 4 = 12 - 3x$ for $x$. 2. **Formula and rules:** Same as above, isolate $x$ by moving terms and simplifying. 3. **Step 1:** Add $3x$ to both sides to get all $x$ terms on one side: $$5x - 4 + 3x = 12 - 3x + 3x \implies 8x - 4 = 12$$ 4. **Step 2:** Add $4$ to both sides to move constants: $$8x - 4 + 4 = 12 + 4 \implies 8x = 16$$ 5. **Step 3:** Divide both sides by $8$: $$\frac{8x}{\cancel{8}} = \frac{16}{\cancel{8}} \implies x = 2$$ 6. **Final answer:** $x = 2$. **Summary:** Show each step clearly by moving terms, canceling common factors with \cancel{}, and simplifying until $x$ is isolated.