SOLUTION: Kuta Software Infinite Algebra 1: Systems of Equations ... - Free Printable

You've uploaded a worksheet titled “Systems of Equations Word Problems” from *Kuta Software - Infinite Algebra 1*. The student has already written answers and some work for problems 1–4. I’ll go through each problem, verify the correctness of the answers, and provide a clear, step-by-step explanation for each.

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

Student’s Answer: 8 & 4

✔ Correct!

Explanation:

Let the two numbers be x and y.

We are told:
- x + y = 12 → (Equation 1)
- x - y = 4 → (Equation 2)

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

Plug into Equation 1:
8 + y = 12 → y = 4

✔ So the numbers are 8 and 4.

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

Student’s Answer: 8 & 5

✔ Correct!

Explanation:

Let the two numbers be x and y, with x > y.

Given:
- x - y = 3 → (Equation 1)
- x + y = 13 → (Equation 2)

Add the two equations:
(x - y) + (x + y) = 3 + 13
→ 2x = 16
→ x = 8

Plug into Equation 2:
8 + y = 13 → y = 5

✔ So the numbers are 8 and 5.

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## ✔ Problem 3:
> Flying to Kampala with a tailwind a plane averaged 158 km/h. On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air.

Student’s Answer:

- Wind: 23 km/h
- Plane: 135 km/h

✔ Correct!

Explanation:

Let:
- p = speed of plane in still air (km/h)
- w = speed of wind (km/h)

With tailwind: plane speed = p + w = 158 → (Equation 1)
Against wind: plane speed = p - w = 112 → (Equation 2)

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

Plug into Equation 1:
135 + w = 158 → w = 23

✔ So, plane speed = 135 km/h, wind speed = 23 km/h.

*(Student’s handwritten steps match this — well done!)*

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## ✘ Problem 4:
> The school that Stefan goes to is selling tickets to a choral performance. On the first day... sold 3 senior citizen tickets and 1 child ticket for $38. Second day... 3 senior and 2 child tickets for $52. Find prices.

Student’s Answer:

- Senior: $8
- Child: $14

✘ Incorrect!

Let’s check:

If senior = $8, child = $14:

Day 1: 3(8) + 1(14) = 24 + 14 = $38 ✔
Day 2: 3(8) + 2(14) = 24 + 28 = $52 ✔

Wait — that actually adds up correctly!

But let’s solve it properly to verify.

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Correct Solution:

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

Equations:
1) 3s + c = 38
2) 3s + 2c = 52

Subtract Equation 1 from Equation 2:

(3s + 2c) - (3s + c) = 52 - 38
→ c = 14

Now plug into Equation 1:
3s + 14 = 38
→ 3s = 24
→ s = 8

✔ So the student’s answer is actually correct:
Senior ticket = $8, Child ticket = $14

*(The student’s scratch work shows they substituted s=8 into 3s + c = 38 and got c=14 — which is valid. Maybe you thought it was wrong because $14 seems high for a child ticket? But mathematically, it’s correct.)*

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## 🎯 Final Summary:

| Problem | Student’s Answer | Correct? | Notes |
|---------|--------------------------|--------|-------|
| 1 | 8 & 4 | ✔ Yes | Perfect |
| 2 | 8 & 5 | ✔ Yes | Perfect |
| 3 | Wind: 23, Plane: 135 | ✔ Yes | Well solved |
| 4 | Senior: $8, Child: $14 | ✔ Yes | Math checks out — even if price seems high |

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## 💡 Tip for Student:
You did great on all problems! For word problems, always:
1. Define variables clearly.
2. Write two equations based on the story.
3. Solve using substitution or elimination.
4. Check your answer by plugging back into the original problem.

Keep up the good work! 👏

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