Worksheet - Van Ripers Math Class - Free Printable

Absolutely! Let’s solve each of these system of equation word problems step by step, following the 6-step method provided.

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Problem 1: Paul, the magician is thinking of two numbers. The sum of two numbers is 36. Their difference is 24. Find the numbers.

Step 1: Let Statement
> Let x = the larger number
> Let y = the smaller number

Step 2: Write 2 equations
> x + y = 36 (sum is 36)
> x - y = 24 (difference is 24)

Step 3: Solve for 1 variable
Add the two equations to eliminate y:

```
x + y = 36
x - y = 24
-----------
2x = 60
→ x = 30
```

Step 4: Solve for other variable
Substitute x = 30 into first equation:

30 + y = 36 → y = 6

Step 5: Answer in Let Statement
> The larger number is 30 and the smaller number is 6.

Step 6: Check
- Sum: 30 + 6 = 36 ✔
- Difference: 30 - 6 = 24 ✔

✔ Final Answer: 30 and 6

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Problem 2: The sum of two numbers is 18. Two times the larger plus 3 times the smaller equals 27. Find the numbers.

Step 1: Let Statement
> Let x = the larger number
> Let y = the smaller number

Step 2: Write 2 equations
> x + y = 18 (sum is 18)
> 2x + 3y = 27 (twice larger + three times smaller = 27)

Step 3: Solve for 1 variable
From first equation: x = 18 - y

Substitute into second equation:

2(18 - y) + 3y = 27
36 - 2y + 3y = 27
36 + y = 27
y = 27 - 36
→ y = -9

Step 4: Solve for other variable
x = 18 - (-9) = 18 + 9 = 27

Step 5: Answer in Let Statement
> The larger number is 27 and the smaller number is -9.

Step 6: Check
- Sum: 27 + (-9) = 18 ✔
- 2x + 3y = 2(27) + 3(-9) = 54 - 27 = 27 ✔

✔ Final Answer: 27 and -9

*(Note: Negative numbers are acceptable unless specified otherwise.)*

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Problem 3: A warehouse stacks 3 large boxes and 2 small boxes to a height of 11 feet. It also stacks 2 large boxes and 1 small box to a height of 7 feet. What are the heights of a large and small box?

Step 1: Let Statement
> Let x = height of one large box (in feet)
> Let y = height of one small box (in feet)

Step 2: Write 2 equations
> 3x + 2y = 11 (3 large + 2 small = 11 ft)
> 2x + y = 7 (2 large + 1 small = 7 ft)

Step 3: Solve for 1 variable
Use substitution or elimination. Let’s use elimination.

Multiply second equation by 2 to match y-coefficients:

2*(2x + y) = 2*7 → 4x + 2y = 14

Now subtract first equation from this:

```
4x + 2y = 14
3x + 2y = 11
-----------
x = 3
```

→ x = 3

Step 4: Solve for other variable
Plug x = 3 into 2x + y = 7:

2(3) + y = 7 → 6 + y = 7 → y = 1

Step 5: Answer in Let Statement
> The height of a large box is 3 feet and the height of a small box is 1 foot.

Step 6: Check
- 3(3) + 2(1) = 9 + 2 = 11 ✔
- 2(3) + 1 = 6 + 1 = 7 ✔

✔ Final Answer: Large box = 3 ft, Small box = 1 ft

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Problem 4: At a store, 3 notebooks and 2 pencils cost $2.80. At the same prices, 2 notebooks and 5 pencils cost $2.60. Find the cost of one notebook and one pencil.

Step 1: Let Statement
> Let x = cost of one notebook (in dollars)
> Let y = cost of one pencil (in dollars)

Step 2: Write 2 equations
> 3x + 2y = 2.80 (3 notebooks + 2 pencils)
> 2x + 5y = 2.60 (2 notebooks + 5 pencils)

Step 3: Solve for 1 variable
Let’s eliminate one variable. Multiply first equation by 5 and second by 2 to eliminate y:

Eq1 * 5: 15x + 10y = 14.00
Eq2 * 2: 4x + 10y = 5.20

Now subtract:

```
15x + 10y = 14.00
4x + 10y = 5.20
------------------
11x = 8.80
→ x = 8.80 / 11 = 0.80
```

→ x = 0.80

Step 4: Solve for other variable
Plug x = 0.80 into 3x + 2y = 2.80:

3(0.80) + 2y = 2.80 → 2.40 + 2y = 2.80 → 2y = 0.40 → y = 0.20

Step 5: Answer in Let Statement
> One notebook costs $0.80 and one pencil costs $0.20.

Step 6: Check
- 3(0.80) + 2(0.20) = 2.40 + 0.40 = 2.80 ✔
- 2(0.80) + 5(0.20) = 1.60 + 1.00 = 2.60 ✔

✔ Final Answer: Notebook = $0.80, Pencil = $0.20

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## ✔ Final Answers Summary:

| Problem | Answer |
|---------|--------|
| 1 | Larger number: 30, Smaller number: 6 |
| 2 | Larger number: 27, Smaller number: -9 |
| 3 | Large box: 3 ft, Small box: 1 ft |
| 4 | Notebook: $0.80, Pencil: $0.20 |

Let me know if you’d like to see graphs or alternate methods (like substitution vs elimination)!