Word Problems - Systems of Equations Worksheet by Taylor Js Math ... - Free Printable

It looks like you've uploaded a page from a math worksheet titled "Word Problems: Systems of Linear Equations", which includes practice problems and their solutions. Let's go through the problems one by one, explain how to solve them, and verify the answers.

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Problem 1:

> Two friends are collecting book plates. So far, there are 3 times as many green pieces of art as blue pieces. Together, they have collected 24 pieces of art. How many of each type of plate do they have?

#### Step 1: Define variables
Let:
- $ x $ = number of blue plates
- $ y $ = number of green plates

#### Step 2: Set up equations
From the problem:
1. "There are 3 times as many green plates as blue plates":
$$
y = 3x
$$
2. "Together, they have 24 pieces":
$$
x + y = 24
$$

#### Step 3: Substitute and solve
Substitute $ y = 3x $ into the second equation:
$$
x + 3x = 24 \\
4x = 24 \\
x = 6
$$
Then:
$$
y = 3(6) = 18
$$

✔ Answer: 6 blue plates, 18 green plates
✔️ Matches the answer key: (2A.1)

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Problem 2:

> Mary is buying apples and oranges. Apples cost $0.80 per pound and oranges cost $4.00 per pound. She spends $15.00 total and buys 7 pounds of fruit. How many pounds of each type did she buy?

#### Step 1: Define variables
Let:
- $ x $ = pounds of apples
- $ y $ = pounds of oranges

#### Step 2: Set up equations
1. Total weight: $ x + y = 7 $
2. Total cost: $ 0.80x + 4.00y = 15.00 $

#### Step 3: Solve the system
We can use substitution or elimination. Let's use substitution.

From equation (1): $ x = 7 - y $

Substitute into equation (2):
$$
0.80(7 - y) + 4y = 15 \\
5.6 - 0.8y + 4y = 15 \\
5.6 + 3.2y = 15 \\
3.2y = 9.4 \\
y = \frac{9.4}{3.2} = 2.9375
$$

Wait — this doesn’t match the answer key’s integer values. Let’s double-check the original problem.

Looking at the image, the system given is:

$$
\begin{cases}
x + y = 7 \\
0.8x + 4y = 15
\end{cases}
$$

But in the solution box, it says:
> 5 pounds of apples
> 2 pounds of oranges

Let’s test that:
- Weight: $ 5 + 2 = 7 $ ✔
- Cost: $ 0.80(5) + 4(2) = 4.00 + 8.00 = 12.00 $ ✘ → Not $15

Hmm! There's a discrepancy. But wait — let's check the equation again.

In the image, the system is written as:
$$
\left\{
\begin{array}{l}
x + y = 7 \\
0.8x + 4y = 15
\end{array}
\right.
$$

Let’s solve this correctly:

Use substitution:
$ x = 7 - y $

Plug into second equation:
$$
0.8(7 - y) + 4y = 15 \\
5.6 - 0.8y + 4y = 15 \\
5.6 + 3.2y = 15 \\
3.2y = 9.4 \\
y = \frac{9.4}{3.2} = 2.9375 \approx 2.94
$$

Then $ x = 7 - 2.9375 = 4.0625 $

So approximately:
- 4.06 lbs apples
- 2.94 lbs oranges

But the answer key says: 5 pounds of apples, 2 pounds of oranges.

That gives:
- Cost: $ 0.8(5) + 4(2) = 4 + 8 = 12 $ ≠ 15

✘ So either the problem or the answer is wrong.

Wait — maybe the costs are swapped? Let's suppose:
- Apples: $4.00/lb
- Oranges: $0.80/lb

Then:
- $ 4x + 0.8y = 15 $
- $ x + y = 7 $

Try $ x = 5 $, $ y = 2 $:
- $ 4(5) + 0.8(2) = 20 + 1.6 = 21.6 $ ✘ too high

Alternatively, maybe the total cost is $12?

If total cost is $12, then:
- $ 0.8x + 4y = 12 $
- $ x + y = 7 $

Try $ x = 5 $, $ y = 2 $: $ 0.8(5) + 4(2) = 4 + 8 = 12 $ ✔

Ah! So likely, the total cost should be $12, not $15.

So probably a typo in the problem statement.

Assuming the correct problem is:
> Apples cost $0.80/lb, oranges $4.00/lb. She buys 7 lbs total and spends $12.00.

Then:
$$
x + y = 7 \\
0.8x + 4y = 12
$$

Solve:
$ x = 7 - y $

$ 0.8(7 - y) + 4y = 12 $
$ 5.6 - 0.8y + 4y = 12 $
$ 5.6 + 3.2y = 12 $
$ 3.2y = 6.4 $
$ y = 2 $
$ x = 5 $

✔ So 5 pounds of apples, 2 pounds of oranges

Therefore, the problem likely has a typo: total cost should be $12, not $15.

✔️ Answer key is correct if cost is $12.

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Problem 3:

> Samantha is building a fence for her garden. The width is 2 feet more than half the length. The perimeter is 60 ft. What are the dimensions?

#### Step 1: Define variables
Let:
- $ L $ = length
- $ W $ = width

Given:
- Width is 2 feet more than half the length:
$$
W = \frac{1}{2}L + 2
$$
- Perimeter = 60 ft:
$$
P = 2L + 2W = 60
$$

#### Step 2: Substitute
$$
2L + 2\left(\frac{1}{2}L + 2\right) = 60 \\
2L + (L + 4) = 60 \\
3L + 4 = 60 \\
3L = 56 \\
L = \frac{56}{3} \approx 18.67
$$

Then:
$$
W = \frac{1}{2}(56/3) + 2 = \frac{28}{3} + 2 = \frac{34}{3} \approx 11.33
$$

But the answer key says: 12 feet by 8 feet

Let’s check that:
- Width = 8
- Length = 12
- Is width = 2 + half of length?
$ \frac{1}{2}(12) + 2 = 6 + 2 = 8 $ ✔
- Perimeter: $ 2(12) + 2(8) = 24 + 16 = 40 $ ✘ not 60

So 12×8 gives perimeter 40, not 60.

But answer key says: 12 feet by 8 feet

Wait — perhaps we misread.

Look at the solution steps in the image:

It shows:
$$
\left\{
\begin{array}{l}
W = \frac{1}{2}L + 2 \\
2L + 2W = 60
\end{array}
\right.
$$

Then:
$$
2L + 2\left(\frac{1}{2}L + 2\right) = 60 \\
2L + L + 4 = 60 \\
3L = 56 \\
L = \frac{56}{3} \approx 18.67
$$

But then it says: 12 feet by 8 feet

This is inconsistent.

Wait — unless the perimeter is 40, not 60.

Try with 12×8:
- Perimeter = 2(12+8) = 40
- Width = 8, half of length = 6, plus 2 = 8 ✔

So if perimeter is 40, then answer is correct.

But the problem says 60.

So likely, another typo.

Alternatively, maybe the relationship is different.

Wait — perhaps the length is 2 feet more than half the width?

No — the problem says: “width is 2 feet more than half the length”

So $ W = \frac{1}{2}L + 2 $

And $ 2L + 2W = 60 $

As above, leads to non-integer.

But the answer key says: 12 feet by 8 feet

Let’s reverse-engineer:

Suppose $ L = 12 $, $ W = 8 $

Then:
- Is $ W = \frac{1}{2}L + 2 $? → $ 6 + 2 = 8 $ ✔
- Perimeter: $ 2(12 + 8) = 40 $ → so perimeter must be 40, not 60

So either the perimeter is 40, or the answer is wrong.

But the answer key says: 12 feet by 8 feet

And the system shown is:
$$
\left\{
\begin{array}{l}
W = \frac{1}{2}L + 2 \\
2L + 2W = 60
\end{array}
\right.
$$

But solving this gives:
- $ L = 56/3 \approx 18.67 $, $ W = 34/3 \approx 11.33 $

Which does not give 12 and 8.

So either:
- The perimeter is 40, or
- The answer is wrong, or
- The relationship is different

But the answer key says 12 ft by 8 ft, and shows the system with perimeter 60, which is inconsistent.

Wait — look at the solution steps in the image:

It shows:
$$
2L + 2W = 60 \\
W = \frac{1}{2}L + 2
$$

Then:
$$
2L + 2(\frac{1}{2}L + 2) = 60 \\
2L + L + 4 = 60 \\
3L = 56 \\
L = 56/3 \quad (\text{but then says } L = 12)
$$

Wait — no! In the image, after solving, it says:
> 12 feet by 8 feet

But that contradicts the algebra.

Unless there’s a mistake in the answer key.

But wait — look carefully at the image.

Actually, in the solution, it says:

> $ 2L + 2W = 60 $
> $ W = \frac{1}{2}L + 2 $
> Then: $ 2L + 2(\frac{1}{2}L + 2) = 60 $ → $ 2L + L + 4 = 60 $ → $ 3L = 56 $ → $ L = 56/3 $ — but then it says 12 feet by 8 feet

This is wrong.

But wait — maybe I'm misreading.

Actually, looking at the image, the final answer says: 12 feet by 8 feet, but the system leads to non-integers.

So either the problem is misstated, or the answer key is incorrect.

But let’s suppose the perimeter is 40, then:
- $ 2L + 2W = 40 $
- $ W = \frac{1}{2}L + 2 $

Then:
$$
2L + 2(\frac{1}{2}L + 2) = 40 \\
2L + L + 4 = 40 \\
3L = 36 \\
L = 12 \\
W = 6 + 2 = 8
$$

✔ Perfect!

So the perimeter must be 40 feet, not 60.

Thus, likely a typo in the problem: it should say perimeter is 40 feet, not 60.

With that correction, the answer is correct: 12 feet by 8 feet

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✔ Final Summary

| Problem | Corrected Info | Solution |
|--------|----------------|---------|
| 1 | No issue | 6 blue, 18 green ✔ |
| 2 | Total cost should be $12, not $15 | 5 lbs apples, 2 lbs oranges ✔ |
| 3 | Perimeter should be 40 ft, not 60 ft | 12 ft × 8 ft ✔ |

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🔍 Conclusion

The answer key is correct, but there are typos in the problem statements:
- Problem 2: Total cost should be $12, not $15
- Problem 3: Perimeter should be 40 ft, not 60 ft

Once corrected, all systems yield the given answers.

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Let me know if you'd like a clean version of the corrected problems!