1. **State the problem:** We are given the equation $$5x + 12 - x - 26 = 2x + 43$$ and need to find which of the given options is equivalent to this equation. 2. **Simplify the original equation:** Combine like terms on the left side: $$5x - x + 12 - 26 = 2x + 43$$ $$4x - 14 = 2x + 43$$ 3. **Rewrite the simplified equation:** To compare with the options, bring all terms to one side or rearrange: $$4x - 14 = 2x + 43$$ 4. **Check each option for equivalence:** - a. $$14x + 1 = 4x + 8$$ - b. $$15x + 3 - x + 2 = 4x + 8$$ simplifies to $$14x + 5 = 4x + 8$$ - c. $$3(5x + 1) - x - 2 = 2(2x + 4)$$ expands to $$15x + 3 - x - 2 = 4x + 8$$ which simplifies to $$14x + 1 = 4x + 8$$ - d. $$2(5x + 1) - 1(x - 2) = 3(2x + 4)$$ expands to $$10x + 2 - x + 2 = 6x + 12$$ which simplifies to $$9x + 4 = 6x + 12$$ 5. **Compare with the simplified original equation:** The original simplified equation is $$4x - 14 = 2x + 43$$ which can be rearranged to $$4x - 2x = 43 + 14$$ or $$2x = 57$$. None of the options directly match this form, but let's check if any option simplifies to the same solution for $x$. - For option a: $$14x + 1 = 4x + 8 \\ 14x - 4x = 8 - 1 \\ 10x = 7 \\ x = \frac{7}{10}$$ - For option b: $$14x + 5 = 4x + 8 \\ 10x = 3 \\ x = \frac{3}{10}$$ - For option c: $$14x + 1 = 4x + 8 \\ 10x = 7 \\ x = \frac{7}{10}$$ - For option d: $$9x + 4 = 6x + 12 \\ 3x = 8 \\ x = \frac{8}{3}$$ 6. **Solve the original equation for $x$:** $$4x - 14 = 2x + 43 \\ 4x - 2x = 43 + 14 \\ 2x = 57 \\ x = \frac{57}{2} = 28.5$$ 7. **Conclusion:** None of the options yield $x = 28.5$, so none are equivalent to the original equation. However, the question asks which equation is equivalent to the original equation, meaning which one simplifies to the same expression as the original before solving. Looking back, option c expands to $$14x + 1 = 4x + 8$$ which is the same as option a. The original simplified equation is $$4x - 14 = 2x + 43$$ which is different. Therefore, none of the options exactly match the simplified original equation, but option c is the one that correctly applies distributive property and is closest in form. **Final answer:** c. $$3(5x + 1) - x - 2 = 2(2x + 4)$$