1. Problem: Solve the system of equations for children ($x$) and adults ($y$): $$x + y = 2200$$ $$1.5x + 4y = 5050$$ Step 1: From the first equation, express $y$ in terms of $x$: $$y = 2200 - x$$ Step 2: Substitute $y$ into the second equation: $$1.5x + 4(2200 - x) = 5050$$ Step 3: Expand and simplify: $$1.5x + 8800 - 4x = 5050$$ Step 4: Combine like terms: $$\cancel{1.5x} - \cancel{4x} = -2.5x$$ $$-2.5x + 8800 = 5050$$ Step 5: Subtract 8800 from both sides: $$-2.5x = 5050 - 8800$$ $$-2.5x = -3750$$ Step 6: Divide both sides by -2.5: $$x = \frac{-3750}{-2.5} = 1500$$ Step 7: Find $y$: $$y = 2200 - 1500 = 700$$ Answer: There are 1500 children and 700 adults. --- 2. Problem: Find the measures of the angles in a triangle where the largest angle ($x$) equals the sum of the other two angles, and twice the smallest angle is 10° less than the largest angle. Step 1: Let $x$ = largest angle, $y$ = smaller angle, and $z$ = smallest angle. Step 2: Use the triangle angle sum: $$x + y + z = 180$$ Step 3: Given $x = y + z$, substitute into the sum: $$y + z + y + z = 180$$ $$2y + 2z = 180$$ Step 4: Simplify: $$y + z = 90$$ Step 5: Given $2z = x - 10$, substitute $x = y + z$: $$2z = y + z - 10$$ Step 6: Rearrange: $$2z - z = y - 10$$ $$z = y - 10$$ Step 7: Substitute $z = y - 10$ into $y + z = 90$: $$y + (y - 10) = 90$$ $$2y - 10 = 90$$ Step 8: Solve for $y$: $$2y = 100$$ $$y = 50$$ Step 9: Find $z$: $$z = 50 - 10 = 40$$ Step 10: Find $x$: $$x = y + z = 50 + 40 = 90$$ Answer: The angles are 90°, 50°, and 40°. --- 3. Problem: A man invests in two accounts, $x$ in the 6% account and $y$ in the 10% account, with $x = 2y$. Total interest is 3520. Step 1: Write the interest equation: $$0.06x + 0.10y = 3520$$ Step 2: Substitute $x = 2y$: $$0.06(2y) + 0.10y = 3520$$ Step 3: Simplify: $$0.12y + 0.10y = 3520$$ $$0.22y = 3520$$ Step 4: Solve for $y$: $$y = \frac{3520}{0.22} = 16000$$ Step 5: Find $x$: $$x = 2 \times 16000 = 32000$$ Answer: He invested 32000 in the 6% account and 16000 in the 10% account.