Handwritten solutions to algebra word problems on systems of equations, including finding numbers based on sum and difference, calculating plane and wind speeds, and determining ticket prices.

Let’s solve each problem step by step.

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Problem 1:
Find the value of two numbers if their sum is 12 and their difference is 4.

Let the two numbers be x and y.

We can write:
- x + y = 12 (sum)
- x - y = 4 (difference)

Add the two equations together:
(x + y) + (x - y) = 12 + 4
→ 2x = 16
→ x = 8

Now plug x = 8 into the first equation:
8 + y = 12
→ y = 4

✔ Check: 8 + 4 = 12, 8 - 4 = 4 → Correct!

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Problem 2:
The difference of two numbers is 3. Their sum is 13. Find the numbers.

Let the numbers be x and y.

Equations:
- x - y = 3
- x + y = 13

Add them:
(x - y) + (x + y) = 3 + 13
→ 2x = 16
→ x = 8

Plug into second equation:
8 + y = 13
→ y = 5

✔ Check: 8 - 5 = 3, 8 + 5 = 13 → Correct!

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Problem 3:
Flying to Kampala with a tailwind, plane averaged 158 km/h. On return trip against wind, averaged 112 km/h. Find speed of wind and speed of plane in still air.

Let:
- p = speed of plane in still air
- w = speed of wind

With tailwind: p + w = 158
Against wind: p - w = 112

Add the two equations:
(p + w) + (p - w) = 158 + 112
→ 2p = 270
→ p = 135

Now plug into first equation:
135 + w = 158
→ w = 23

✔ Check: 135 + 23 = 158, 135 - 23 = 112 → Correct!

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Problem 4:
School sold tickets:
- Day 1: 3 senior + 1 child = $38
- Day 2: 3 senior + 2 child = $52

Let:
- s = price of senior ticket
- c = price of child ticket

Equations:
- 3s + c = 38
- 3s + 2c = 52

Subtract first equation from second:
(3s + 2c) - (3s + c) = 52 - 38
→ c = 14

Plug c = 14 into first equation:
3s + 14 = 38
→ 3s = 24
→ s = 8

✔ Check: 3×8 + 14 = 24 + 14 = 38; 3×8 + 2×14 = 24 + 28 = 52 → Correct!

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Final Answer:

1) The two numbers are 8 and 4.
2) The two numbers are 8 and 5.
3) Speed of wind: 23 km/h, speed of plane in still air: 135 km/h.
4) Senior citizen ticket: $8, child ticket: $14.