Solve systems of linear equations to crack the code in this engaging math activity.

Let's solve each system of linear equations step by step and match the solutions to the corresponding letters in the table. Then we'll decode the code using the four boxes at the bottom.

---

Step 1: Solve Each System

---

#### 1.
$$
\begin{align*}
3x + 2y &= 9 \quad \text{(1)}\\
x + 2y &= 7 \quad \text{(2)}
\end{align*}
$$

Subtract (2) from (1):
$$
(3x + 2y) - (x + 2y) = 9 - 7 \Rightarrow 2x = 2 \Rightarrow x = 1
$$

Plug into (2):
$$
1 + 2y = 7 \Rightarrow 2y = 6 \Rightarrow y = 3
$$

So, $ x = 1, y = 3 $ → Look for this in the table.

→ I: $ x=1, y=6 $? No
→ E: $ x=1, y=9 $? No
→ S: $ x=1, y=3 $? Yes! ✔

So, 1 → S

---

#### 2.
$$
\begin{align*}
5x + y &= 14 \quad \text{(1)}\\
3x + y &= 10 \quad \text{(2)}
\end{align*}
$$

Subtract (2) from (1):
$$
(5x + y) - (3x + y) = 14 - 10 \Rightarrow 2x = 4 \Rightarrow x = 2
$$

Plug into (2):
$$
3(2) + y = 10 \Rightarrow 6 + y = 10 \Rightarrow y = 4
$$

So, $ x = 2, y = 4 $

→ C: $ x=2, y=7 $? No
→ D: $ x=2, y=4 $? Yes! ✔

So, 2 → D

---

#### 3.
$$
\begin{align*}
9x + 5y &= 50 \quad \text{(1)}\\
12x + 5y &= 65 \quad \text{(2)}
\end{align*}
$$

Subtract (1) from (2):
$$
(12x + 5y) - (9x + 5y) = 65 - 50 \Rightarrow 3x = 15 \Rightarrow x = 5
$$

Plug into (1):
$$
9(5) + 5y = 50 \Rightarrow 45 + 5y = 50 \Rightarrow 5y = 5 \Rightarrow y = 1
$$

So, $ x = 5, y = 1 $

→ K: $ x=5, y=1 $? Yes! ✔

So, 3 → K

---

#### 4.
$$
\begin{align*}
4x + y &= 37 \quad \text{(1)}\\
2x - y &= 17 \quad \text{(2)}
\end{align*}
$$

Add (1) and (2):
$$
(4x + y) + (2x - y) = 37 + 17 \Rightarrow 6x = 54 \Rightarrow x = 9
$$

Plug into (1):
$$
4(9) + y = 37 \Rightarrow 36 + y = 37 \Rightarrow y = 1
$$

So, $ x = 9, y = 1 $

→ B: $ x=9, y=1 $? Yes! ✔

So, 4 → B

---

#### 5.
$$
\begin{align*}
x + 2y &= 22 \quad \text{(1)}\\
2x + y &= 20 \quad \text{(2)}
\end{align*}
$$

Multiply (1) by 2:
$$
2x + 4y = 44 \quad \text{(3)}
$$

Now subtract (2) from (3):
$$
(2x + 4y) - (2x + y) = 44 - 20 \Rightarrow 3y = 24 \Rightarrow y = 8
$$

Plug into (1):
$$
x + 2(8) = 22 \Rightarrow x + 16 = 22 \Rightarrow x = 6
$$

So, $ x = 6, y = 8 $

→ Z: $ x=6, y=8 $? Yes! ✔

So, 5 → Z

---

#### 6.
$$
\begin{align*}
x - y &= -1 \quad \text{(1)}\\
2x + y &= 4 \quad \text{(2)}
\end{align*}
$$

Add (1) and (2):
$$
(x - y) + (2x + y) = -1 + 4 \Rightarrow 3x = 3 \Rightarrow x = 1
$$

Plug into (1):
$$
1 - y = -1 \Rightarrow y = 2
$$

So, $ x = 1, y = 2 $

→ L: $ x=1, y=2 $? Yes! ✔

So, 6 → L

---

#### 7.
$$
\begin{align*}
2x - y &= 14 \quad \text{(1)}\\
5x + y &= 14 \quad \text{(2)}
\end{align*}
$$

Add (1) and (2):
$$
(2x - y) + (5x + y) = 14 + 14 \Rightarrow 7x = 28 \Rightarrow x = 4
$$

Plug into (1):
$$
2(4) - y = 14 \Rightarrow 8 - y = 14 \Rightarrow y = -6
$$

So, $ x = 4, y = -6 $

→ T: $ x=4, y=-6 $? Yes! ✔

So, 7 → T

---

#### 8.
$$
\begin{align*}
x + y &= -7 \quad \text{(1)}\\
x - y &= -3 \quad \text{(2)}
\end{align*}
$$

Add (1) and (2):
$$
2x = -10 \Rightarrow x = -5
$$

Plug into (1):
$$
-5 + y = -7 \Rightarrow y = -2
$$

So, $ x = -5, y = -2 $

→ P: $ x=-5, y=-2 $? Yes! ✔

So, 8 → P

---

#### 9.
$$
\begin{align*}
2x + 3y &= 1 \quad \text{(1)}\\
5x + 4y &= -1 \quad \text{(2)}
\end{align*}
$$

Use elimination.

Multiply (1) by 5: $ 10x + 15y = 5 $
Multiply (2) by 2: $ 10x + 8y = -2 $

Subtract:
$$
(10x + 15y) - (10x + 8y) = 5 - (-2) \Rightarrow 7y = 7 \Rightarrow y = 1
$$

Plug into (1):
$$
2x + 3(1) = 1 \Rightarrow 2x + 3 = 1 \Rightarrow 2x = -2 \Rightarrow x = -1
$$

So, $ x = -1, y = 1 $

→ F: $ x=-1, y=1 $? Yes! ✔

So, 9 → F

---

#### 10.
$$
\begin{align*}
3x - 9 &= -4y \quad \text{(1)}\\
2x - y &= -16 \quad \text{(2)}
\end{align*}
$$

Rewrite (1):
$$
3x + 4y = 9 \quad \text{(1')}
$$

Now solve:

(1'): $ 3x + 4y = 9 $
(2): $ 2x - y = -16 $

Solve (2) for $ y $:
$$
y = 2x + 16
$$

Plug into (1'):
$$
3x + 4(2x + 16) = 9 \Rightarrow 3x + 8x + 64 = 9 \Rightarrow 11x = -55 \Rightarrow x = -5
$$

Then $ y = 2(-5) + 16 = -10 + 16 = 6 $

So, $ x = -5, y = 6 $

→ V: $ x=-5, y=6 $? Yes! ✔

So, 10 → V

---

#### 11.
$$
\begin{align*}
6x - 5y &= -32 \quad \text{(1)}\\
5x + 4y &= 6 \quad \text{(2)}
\end{align*}
$$

Use elimination.

Multiply (1) by 4: $ 24x - 20y = -128 $
Multiply (2) by 5: $ 25x + 20y = 30 $

Add:
$$
(24x - 20y) + (25x + 20y) = -128 + 30 \Rightarrow 49x = -98 \Rightarrow x = -2
$$

Plug into (2):
$$
5(-2) + 4y = 6 \Rightarrow -10 + 4y = 6 \Rightarrow 4y = 16 \Rightarrow y = 4
$$

So, $ x = -2, y = 4 $

→ A: $ x=-2, y=4 $? Yes! ✔

So, 11 → A

---

#### 12.
$$
\begin{align*}
6x - 3y + 27 &= 0 \Rightarrow 6x - 3y = -27 \quad \text{(1)}\\
5x + 4y &= 10 \quad \text{(2)}
\end{align*}
$$

Simplify (1): divide by 3 → $ 2x - y = -9 $ → $ y = 2x + 9 $

Plug into (2):
$$
5x + 4(2x + 9) = 10 \Rightarrow 5x + 8x + 36 = 10 \Rightarrow 13x = -26 \Rightarrow x = -2
$$

Then $ y = 2(-2) + 9 = -4 + 9 = 5 $

So, $ x = -2, y = 5 $

→ Check table:
Is there $ x = -2, y = 5 $?
→ R: $ x=-2, y=5 $? Yes! ✔

So, 12 → R

---

Summary of Answers:

| Problem | Solution | Letter |
|--------|----------|--------|
| 1 | $ x=1, y=3 $ | S |
| 2 | $ x=2, y=4 $ | D |
| 3 | $ x=5, y=1 $ | K |
| 4 | $ x=9, y=1 $ | B |
| 5 | $ x=6, y=8 $ | Z |
| 6 | $ x=1, y=2 $ | L |
| 7 | $ x=4, y=-6 $ | T |
| 8 | $ x=-5, y=-2 $ | P |
| 9 | $ x=-1, y=1 $ | F |
| 10 | $ x=-5, y=6 $ | V |
| 11 | $ x=-2, y=4 $ | A |
| 12 | $ x=-2, y=5 $ | R |

---

Now fill the code boxes:

The bottom says:

```
[1 - 2 - 3] [4 - 5 - 6] [7 - 8 - 9] [10 - 11 - 12]
```

We need to take the letters corresponding to problems 1–3, 4–6, etc., and form the code.

So:

- 1-2-3: S, D, K → SDK
- 4-5-6: B, Z, L → BZL
- 7-8-9: T, P, F → TPF
- 10-11-12: V, A, R → VAR

But wait — these are codes, not words. But maybe it's a word?

Wait — let’s check if it makes sense.

But perhaps the format is just to write the letters in order.

But looking back: the puzzle says “complete the code in the four boxes”.

So likely, we are to write the letters for each group.

So final answer:

- First box: S D K → SDK
- Second: B Z L → BZL
- Third: T P F → TPF
- Fourth: V A R → VAR

But let’s double-check:

Wait — actually, the image shows the boxes as:

```
[1 - 2 - 3] [4 - 5 - 6] [7 - 8 - 9] [10 - 11 - 12]
```

And the lock suggests a code.

But "SDK BZL TPF VAR" doesn’t look like a word.

Wait — maybe we are supposed to read the letters in sequence?

Let’s list all the letters in order:

1. S
2. D
3. K
4. B
5. Z
6. L
7. T
8. P
9. F
10. V
11. A
12. R

So full string: S D K B Z L T P F V A R

That seems random.

Wait — but perhaps we made a mistake?

Let me recheck problem 1:

We had:
1. $ x=1, y=3 $ → S? Let's confirm:

Look at table:

- S: $ x=1, y=3 $? Yes → Correct.

2. $ x=2, y=4 $ → D? D: $ x=2, y=4 $? Yes.

3. $ x=5, y=1 $ → K? K: $ x=5, y=1 $? Yes.

4. $ x=9, y=1 $ → B? B: $ x=9, y=1 $? Yes.

5. $ x=6, y=8 $ → Z? Z: $ x=6, y=8 $? Yes.

6. $ x=1, y=2 $ → L? L: $ x=1, y=2 $? Yes.

7. $ x=4, y=-6 $ → T? T: $ x=4, y=-6 $? Yes.

8. $ x=-5, y=-2 $ → P? P: $ x=-5, y=-2 $? Yes.

9. $ x=-1, y=1 $ → F? F: $ x=-1, y=1 $? Yes.

10. $ x=-5, y=6 $ → V? V: $ x=-5, y=6 $? Yes.

11. $ x=-2, y=4 $ → A? A: $ x=-2, y=4 $? Yes.

12. $ x=-2, y=5 $ → R? R: $ x=-2, y=5 $? Yes.

All correct.

Now, maybe the "code" is just the sequence of letters in the boxes.

But perhaps the code is meant to be read as:

- Box 1: S D K → SDK
- Box 2: B Z L → BZL
- Box 3: T P F → TPF
- Box 4: V A R → VAR

But that doesn't make a word.

Wait — could it be that the boxes are to be filled with the letters, and the final answer is the sequence?

But maybe it's intended to be a word.

Wait — look at 10, 11, 12: V, A, R → VAR? That’s close to "var", but not meaningful.

Wait — what if we look at the solution values?

Alternatively, maybe the code is just the letters, and we’re done.

But let’s try reading them in order:

S D K B Z L T P F V A R

No obvious word.

Wait — perhaps I made a mistake in problem 1?

Wait — problem 1: $ x=1, y=3 $

Table:

- E: $ x=1, y=9 $
- S: $ x=1, y=3 $ → Yes.

But is there a typo?

Wait — maybe the code is not a word, but just the letters.

But perhaps the puzzle expects us to write the letters in the boxes.

So the final answer is:

- First box (1-3): S, D, K
- Second box (4-6): B, Z, L
- Third box (7-9): T, P, F
- Fourth box (10-12): V, A, R

So the completed code is:

SDK – BZL – TPF – VAR

But since the boxes are blank, and the puzzle says “complete the code”, likely the answer is:

> SDK BZL TPF VAR

But perhaps it's meant to be written in the boxes.

Alternatively, maybe the code is something else.

Wait — another thought: perhaps the letters correspond to positions, and we're to spell a word.

But let’s try to see if any of the combinations form words.

SDK? Not a word.

BZL? No.

TPF? No.

VAR? Yes — “var” is a programming keyword, but unlikely.

Wait — perhaps I made a mistake in problem 12?

Problem 12:

Equations:
- $ 6x - 3y + 27 = 0 $ → $ 6x - 3y = -27 $ → divide by 3: $ 2x - y = -9 $
- $ 5x + 4y = 10 $

From first: $ y = 2x + 9 $

Plug into second:
$ 5x + 4(2x + 9) = 10 $
$ 5x + 8x + 36 = 10 $
$ 13x = -26 $ → $ x = -2 $
$ y = 2(-2) + 9 = -4 + 9 = 5 $

So $ x = -2, y = 5 $

Now check table:

- R: $ x = -2, y = 5 $? Yes → R

Correct.

Wait — what about problem 9?

We had $ x = -1, y = 1 $ → F: $ x=-1, y=1 $? Yes.

But is there a better match?

Wait — Q: $ x=-4, y=9 $? No.

F is correct.

Another idea: maybe the code is not the letters, but the values?

But no — the instructions say: “link them to the table above to complete the code”

And the table maps (x,y) to letters.

So we must use the letters.

Perhaps the final code is SDK BZL TPF VAR, but maybe it's meant to be read as a phrase.

Wait — S D K → "SDK"? Like software development kit? Unlikely.

Wait — what if the order is wrong?

Wait — the boxes are labeled:

[1 - 2 - 3] [4 - 5 - 6] [7 - 8 - 9] [10 - 11 - 12]

So we fill:

- First box: S D K
- Second: B Z L
- Third: T P F
- Fourth: V A R

So the final code is:

> SDK BZL TPF VAR

But perhaps the puzzle has a typo, or we're missing something.

Wait — let's check problem 5 again.

Problem 5:

$$
x + 2y = 22 \\
2x + y = 20
$$

We solved: $ x=6, y=8 $

Check:
$ 6 + 2(8) = 6+16=22 $ ✓
$ 2(6)+8=12+8=20 $ ✓

Z: $ x=6, y=8 $? Yes.

But Z is listed as $ x=6, y=8 $? Yes.

All correct.

Wait — perhaps the code is not the letters, but the positions?

No — the table assigns letters to (x,y).

Maybe the code is S D K B Z L T P F V A R, and it's a cipher.

But without more context, likely the answer is simply the letters in the boxes.

So the final answer is:

> SDK – BZL – TPF – VAR

But since the puzzle says “complete the code”, and the boxes are blank, we write:

✔ Final Answer:

SDK BZL TPF VAR

Or, in boxed form as per the image:

```
[ S D K ] [ B Z L ] [ T P F ] [ V A R ]
```

So the decoded code is:

> SDK BZL TPF VAR

Even though it's not a real word, it's the solution based on the matching.

---

✔ Conclusion:

After solving all systems and matching to the table, the code is:

SDK – BZL – TPF – VAR

This is the completed code for the lock.