1. **State the problem:** Solve the system of linear equations: $$3a + 6b = 45$$ $$2a - 2b = 12$$ Find the values of $a$ and $b$ that satisfy both equations. 2. **Use the linear combination method:** Multiply equations to align coefficients for elimination. 3. Multiply the second equation by 3 to align $a$ coefficients: $$3 \times (2a - 2b) = 3 \times 12$$ $$6a - 6b = 36$$ 4. Multiply the first equation by 2 to align $a$ coefficients: $$2 \times (3a + 6b) = 2 \times 45$$ $$6a + 12b = 90$$ 5. Subtract the second new equation from the first new equation to eliminate $a$: $$\cancel{6a} + 12b - (\cancel{6a} - 6b) = 90 - 36$$ $$12b + 6b = 54$$ $$18b = 54$$ 6. Solve for $b$: $$b = \frac{54}{18} = 3$$ 7. Substitute $b=3$ into the second original equation: $$2a - 2(3) = 12$$ $$2a - 6 = 12$$ 8. Solve for $a$: $$2a = 12 + 6 = 18$$ $$a = \frac{18}{2} = 9$$ 9. **Final solution:** $$(a, b) = (9, 3)$$ 10. Check the solution against the given options: none match $(9,3)$ exactly, so re-check calculations. **Re-examine step 5:** Instead of subtracting, try adding equations to eliminate $b$. Multiply second equation by 3: $$6a - 6b = 36$$ Add to first equation multiplied by 1: $$3a + 6b = 45$$ Adding: $$6a - 6b + 3a + 6b = 36 + 45$$ $$9a = 81$$ $$a = 9$$ Substitute $a=9$ into second equation: $$2(9) - 2b = 12$$ $$18 - 2b = 12$$ $$-2b = -6$$ $$b = 3$$ Solution is $(9,3)$ which is not in the options. Check if options are correct or if problem has a typo. **Alternatively, test options:** - For $(1,7)$: $3(1)+6(7)=3+42=45$ correct; $2(1)-2(7)=2-14=-12$ not 12. - For $(-27,6)$: $3(-27)+6(6)=-81+36=-45$ not 45. - For $(27,-6)$: $3(27)+6(-6)=81-36=45$ correct; $2(27)-2(-6)=54+12=66$ not 12. - For $(-1,7)$: $3(-1)+6(7)=-3+42=39$ not 45. No option matches both equations. **Conclusion:** The solution to the system is $(9,3)$, which is not among the given options.