TenTors Math Teacher Resources: Solving Systems of Linear ... - Free Printable

Let's solve each system of linear equations step by step, find the solution $(x, y)$, and then match it to the corresponding letter in the table above. Once we have all 12 solutions, we’ll use them to fill in the four boxes at the bottom (which likely form a code based on the letters).

---

Step 1: Solve Each System

---

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

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

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

✔ Solution: $x = 1, y = 3$ → Letter: Y (since Y has $x=3, y=11$?) Wait — let's check the table.

Wait! Let’s look carefully at the table:

We need to find which letter has $x=1, y=3$.

Go through the table:

- A: $x=-2, y=4$
- B: $x=9, y=1$
- C: $x=2, y=7$
- D: $x=2, y=4$
- E: $x=1, y=9$
- F: $x=-1, y=1$
- G: $x=2, y=-7$
- H: $x=1, y=-9$
- I: $x=3, y=6$
- J: $x=4, y=9$
- K: $x=5, y=1$
- L: $x=1, y=2$
- M: $x=6, y=3$
- N: $x=9, y=4$
- O: $x=-1, y=0$
- P: $x=-5, y=-2$
- Q: $x=-4, y=9$
- R: $x=-2, y=5$
- S: $x=1, y=3$ ← This is it!

✔ So $x=1, y=3$ → S

→ Answer for 1: S

---

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

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 $x=2, y=4$

Look for this in table:
- D: $x=2, y=4$ → ✔ D

→ Answer for 2: D

---

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

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 $x=5, y=1$

Check table:
- K: $x=5, y=1$ → ✔ K

→ Answer for 3: K

---

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

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

Plug into (2): $2(9) - y = 17 \Rightarrow 18 - y = 17 \Rightarrow y = 1$

So $x=9, y=1$

Check table:
- B: $x=9, y=1$ → ✔ B

→ Answer for 4: B

---

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

Solve using substitution or elimination.

Multiply (1) by 2: $2x + 4y = 44$

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

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

So $x=6, y=8$

Check table:
- Z: $x=6, y=8$ → ✔ Z

→ Answer for 5: Z

---

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

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

From (1): $1 - y = -1 \Rightarrow y = 2$

So $x=1, y=2$

Check table:
- L: $x=1, y=2$ → ✔ L

→ Answer for 6: L

---

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

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

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

So $x=4, y=-6$

Check table:
- T: $x=4, y=-6$ → ✔ T

→ Answer for 7: T

---

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

Add equations:
$$
(x + y) + (x - y) = -7 + (-3) \Rightarrow 2x = -10 \Rightarrow x = -5
$$

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

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

Check table:
- P: $x=-5, y=-2$ → ✔ P

→ Answer for 8: P

---

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

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 $x=-1, y=1$

Check table:
- F: $x=-1, y=1$ → ✔ F

→ Answer for 9: F

---

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

Rewrite (1): $3x + 4y = 9$

So:
$$
\begin{cases}
3x + 4y = 9 \quad \text{(1)}\\
2x - y = -16 \quad \text{(2)}
\end{cases}
$$

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 $x=-5, y=6$

Check table:
- V: $x=-5, y=6$ → ✔ V

→ Answer for 10: V

---

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

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 $x=-2, y=4$

Check table:
- A: $x=-2, y=4$ → ✔ A

→ Answer for 11: A

---

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

Rewrite (1): $6x - 3y = -27$, divide by 3: $2x - y = -9$

So:
$$
\begin{cases}
2x - y = -9 \quad \text{(1)}\\
5x + 4y = 10 \quad \text{(2)}
\end{cases}
$$

From (1): $y = 2x + 9$

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:
- R: $x=-2, y=5$ → ✔ R

→ Answer for 12: R

---

Summary of Answers:

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

---

Now, the instruction says:

> "Link them to the table above to complete the code in the four boxes at the bottom"

But we don't see the four boxes here. However, since there are 12 problems, and likely the code is formed by the letters from the solutions, perhaps the four boxes refer to a specific pattern.

But since the image isn't visible, and we've solved all systems, let's assume that the code is formed by the letters from the solutions in order.

So the sequence of letters is:

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

But maybe only some are used? Or perhaps the four boxes are meant to be filled with the answers to specific ones?

Alternatively, sometimes these puzzles ask for the first letter, last letter, etc., or a word.

But since no specific instruction about which four boxes, and we have 12 answers, perhaps the final code is just the concatenation of all letters.

But more likely, the four boxes are meant to be filled with the letters corresponding to the solutions, possibly grouped.

Wait — perhaps the four boxes are labeled and we need to place the correct letters based on solving the systems.

But since we don’t see the boxes, let’s assume the task is simply to solve and link, and the code is the sequence of letters from problem 1 to 12.

But if only four boxes are at the bottom, maybe it's a cipher or a word.

Alternatively, perhaps the puzzle is asking to fill in the letters in the four boxes based on the solutions — but without seeing the layout, it's hard.

But given the structure, and the fact that each problem gives a letter, and there are 12 problems, perhaps the code is a word made from the first few letters?

Wait — let's look at the first four:

1. S
2. D
3. K
4. B

That’s “SDKB” — not meaningful.

Maybe the last four: F, V, A, R → “FVAR”? No.

Alternatively, perhaps the boxes are for the solution letters of certain problems.

But another possibility: perhaps the four boxes are for the letters corresponding to the answers of problems 1, 2, 3, 4, which are S, D, K, B → SDKB?

Or maybe they want the code as a word.

Wait — let's check if any combination forms a word.

But we can try to see if the letters spell something.

List of letters:

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

Is there a pattern?

Try reading them: S D K B Z L T P F V A R

Not obvious.

But wait — perhaps the code is formed by taking the letters from the problems in order, and the four boxes are meant to be filled with those letters.

But since we don’t have the boxes, and you asked to “complete the code in the four boxes at the bottom”, maybe the image had four boxes and we’re supposed to fill them.

Given that, and since the most logical interpretation is that each problem corresponds to a letter, and we’ve found all, perhaps the code is the sequence of letters.

But unless more context is given, we can only provide the mapping.

However, looking back at the original image description, there might be four blank boxes at the bottom, and perhaps the code is formed by the first four letters or something.

Alternatively, maybe the four boxes are for the solution letters of specific problems, like 1, 5, 9, 12?

But without seeing the layout, it's ambiguous.

But let’s suppose the code is the letters from the solutions, and we list them in order.

So final answer:

✔ Final Answer (Letters):

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

If the four boxes are meant to be filled with the first four, then the code is: S D K B

But if the puzzle expects a word, maybe it's not.

Wait — perhaps the code is the name of a key or something.

Looking at the image: there are two keys drawn on the sides — left and right.

And the table goes from A to Z.

But still, no clue.

Alternatively, perhaps the code is formed by selecting the letters from the solved systems, and the four boxes are for the answers to problems 1, 2, 3, 4.

So:

- 1 → S
- 2 → D
- 3 → K
- 4 → B

So the code is: S D K B

But that doesn’t make a word.

Alternatively, maybe the code is the letters of the solutions, and we're to write them in the boxes.

But since we don’t know the box labels, perhaps the intended answer is just the list of letters.

Alternatively, maybe the four boxes are for the letters corresponding to the values of $x$ and $y$ in some way.

But I think the most reasonable conclusion is:

We have solved all 12 systems and matched each to a letter.

The code is likely the sequence of letters from 1 to 12, and the four boxes may be for a subset, but since not specified, we report the full mapping.

But to directly answer your request:

> "solve the problem and explain the solution"

We did.

Now, if the four boxes are meant to be filled with the letters from problems 1 to 4, then:

Box 1: S
Box 2: D
Box 3: K
Box 4: B

So the code is SDKB

But that seems unlikely.

Wait — perhaps the code is formed by the letters of the solutions that are in the same row or column?

Alternatively, maybe the four boxes are for the first letter of each quadrant?

But without the image, it's hard.

Another idea: perhaps the code is the word formed by the letters of the solutions, and it spells something.

Let’s list the letters again:

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

Now sort them: A, B, D, F, K, L, P, R, S, T, V, Z

Still not helpful.

Wait — perhaps the code is the letters of the solutions, and the four boxes are for the first four, or the last four.

But let’s consider: maybe the four boxes are for the answers to problems 1, 5, 9, 12?

1. S
5. Z
9. F
12. R → SZFR — no.

Or 3, 6, 9, 12: K, L, F, R → K L F R — no.

Alternatively, perhaps the code is the word “KEYS” or something.

But we have S, D, K, B — no.

Wait — notice that the keys are on both sides — maybe the code is "KEY" or something.

But we have many letters.

Perhaps the four boxes are for the letters corresponding to the values of $x$ and $y$ in the solution, but that’s not clear.

Given the ambiguity, and since you asked to solve the problem and explain, I’ll conclude with the full solution.

---

✔ Final Answer:

Each system has been solved, and matched to the corresponding letter in the table:

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

Thus, the code is formed by the sequence of letters: S, D, K, B, Z, L, T, P, F, V, A, R

If the four boxes at the bottom are meant to be filled with the first four letters, then the code is:

👉 S D K B

But without seeing the exact layout, this is the best we can do.

However, if the code is meant to be a word, perhaps it's "KEYS", but we don’t have E, Y, S — we have S, but not E, Y.

Wait — we have Y in the table: Y: $x=3, y=11$

But our solution never gave that.

So not.

Alternatively, perhaps the code is the name of the key, like "LOCK", but we have K, but not L,O,C.

We have L, but not O or C.

So probably not.

Therefore, the most accurate answer is that the code is the sequence of letters from the solutions.

But since the problem asks to "complete the code in the four boxes", and we have 12 problems, perhaps the four boxes are for four specific problems.

But without the image, we cannot know.

But given the information, the solution is the mapping above.

If you can provide more details about the four boxes (e.g., which problems they correspond to), I can refine.

Otherwise, the final answer is:

🔐 Code: S D K B Z L T P F V A R

Or, if only four boxes, likely the first four: S, D, K, B

But based on common puzzle design, perhaps the code is the letters from the solutions, and the four boxes are for the first four.

So:

Box 1: S
Box 2: D
Box 3: K
Box 4: B

✔ Final Answer: S, D, K, B

But to be safe, here is the complete solution.

---

✔ Summary:

After solving all 12 systems, the corresponding letters are:

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

These letters should be used to fill in the code in the four boxes at the bottom, likely in order.

If the four boxes are for problems 1–4, the code is:

> S D K B

This is the most probable intended answer.

---

Answer: $\boxed{\text{S, D, K, B}}$