1. Let x be the number of 1.00 peso coins and y be the number of 5.00 peso coins.
Equations: x + y = 105 and 1x + 5y = 205.
Subtract first equation from second: 4y = 100 → y = 25.
Substitute into first: x + 25 = 105 → x = 80.
Answer: 80 coins of 1.00 peso, 25 coins of 5.00 pesos.
2. Let d be dresses, s be shoes.
Cost: 400d + 1000s = 48000.
Profit: 0.2*400d + 0.5*1000s = 18000 → 80d + 500s = 18000.
Simplify cost: 2d + 5s = 240 (divide by 200).
Simplify profit: 4d + 25s = 900 (divide by 20).
Multiply first simplified by 2: 4d + 10s = 480.
Subtract from second: 15s = 420 → s = 28.
Substitute: 2d + 5*28 = 240 → 2d = 100 → d = 50.
Answer: 50 dresses, 28 shoes.
3. Let x and y be the numbers, x > y.
Equations: x + y = 115, x - y = -21.
Add equations: 2x = 94 → x = 47.
Substitute: 47 + y = 115 → y = 68.
Answer: The numbers are 47 and 68.
4. Let x and y be the angles, x + y = 90.
Assume x is 42 more than half y: x = y/2 + 42.
Substitute: y/2 + 42 + y = 90 → (3/2)y = 48 → y = 32.
Then x = 90 - 32 = 58.
Check: 58 = 32/2 + 42 = 16 + 42 = 58. Correct.
Answer: The angles are 58 degrees and 32 degrees.
5. Let tens digit be t, units digit be u. Number = 10t + u.
Given: 10t + u = 6(t + u) and t = u + 1.
Substitute t: 10(u+1) + u = 6((u+1) + u) → 10u + 10 + u = 6(2u + 1) → 11u + 10 = 12u + 6 → u = 4.
Then t = 5. Number = 10*5 + 4 = 54.
Check: 5+4=9, 6*9=54. Correct.
Answer: The original number is 54.
6. Let x and y be the numbers, assume x = 2y - 7.
Equation: x + y = 29.
Substitute: 2y - 7 + y = 29 → 3y = 36 → y = 12.
Then x = 2*12 - 7 = 17.
Answer: The numbers are 17 and 12.
7. Let h be cost of hamburger, f be cost of french fries.
Equations: 6h + 4f = 330, 4h + 9f = 505.
Multiply first by 2, second by 3: 12h + 8f = 660, 12h + 27f = 1515.
Subtract: 19f = 855 → f = 45.
Substitute: 6h + 4*45 = 330 → 6h = 150 → h = 25.
Answer: Hamburgers cost P25.00, french fries cost P45.00.
8. Let t be number of P20 bills, f be number of P50 bills.
Equations: t + f = 40, 20t + 50f = 1490.
Multiply first by 20: 20t + 20f = 800.
Subtract from second: 30f = 690 → f = 23.
Then t = 40 - 23 = 17.
Answer: 17 P20 bills, 23 P50 bills.
9. Let n be nickels, d be dimes. d = 5n.
Value: 0.05n + 0.10d = 4.40.
Substitute: 0.05n + 0.10*(5n) = 4.40 → 0.05n + 0.50n = 4.40 → 0.55n = 4.40 → n = 8.
Then d = 5*8 = 40.
Answer: 8 nickels, 40 dimes.
10. Let x be liters of 50% solution, y be liters of 20% solution.
Equations: x + y = 12, 0.50x + 0.20y = 0.30*12 = 3.6.
Multiply first by 0.20: 0.20x + 0.20y = 2.4.
Subtract from second: 0.30x = 1.2 → x = 4.
Then y = 12 - 4 = 8.
Answer: 4 liters of 50% solution, 8 liters of 20% solution.
Parent Tip: Review the logic above to help your child master the concept of linear systems word problems worksheet.
