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

Let’s solve each system of linear equations one by one. We’ll find the values of x and y for each, then match them to the table at the top to get the letter code.

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Problem 1:
3x + 2y = 9
x + 2y = 7

Subtract the second equation from the first:
(3x + 2y) - (x + 2y) = 9 - 7 → 2x = 2 → x = 1

Plug x = 1 into second equation:
1 + 2y = 7 → 2y = 6 → y = 3

Look in table: x=1, y=3 → That’s M

Wait — let me double-check the table.

Looking at the table:

- M: x=6, y=3 → no
- E: x=1, y=9 → no
- H: x=1, y=-9 → no
- S: x=1, y=3 → YES! S

So Problem 1 → S

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Problem 2:
5x + y = 14
3x + y = 10

Subtract second from first:
(5x + y) - (3x + y) = 14 - 10 → 2x = 4 → x = 2

Plug into second: 3(2) + y = 10 → 6 + y = 10 → y = 4

Table: x=2, y=4 → Look for that.

C: x=2, y=7 → no
D: x=2, y=4 → YES! D

Problem 2 → D

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Problem 3:
9x + 5y = 50
12x + 5y = 65

Subtract first from second:
(12x + 5y) - (9x + 5y) = 65 - 50 → 3x = 15 → x = 5

Plug into first: 9(5) + 5y = 50 → 45 + 5y = 50 → 5y = 5 → y = 1

Table: x=5, y=1 → K: x=5, y=1 → YES! K

Problem 3 → K

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Problem 4:
4x + y = 37
2x - y = 17

Add the two equations:
(4x + y) + (2x - y) = 37 + 17 → 6x = 54 → x = 9

Plug into second: 2(9) - y = 17 → 18 - y = 17 → y = 1

Table: x=9, y=1 → B: x=9, y=1 → YES! B

Problem 4 → B

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Problem 5:
x + 2y = 22
2x + y = 20

Multiply first equation by 2:
2x + 4y = 44
Now subtract second equation:
(2x + 4y) - (2x + y) = 44 - 20 → 3y = 24 → y = 8

Plug into first: x + 2(8) = 22 → x + 16 = 22 → x = 6

Table: x=6, y=8 → Z: x=6, y=8 → YES! Z

Problem 5 → Z

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Problem 6:
x - y = -1
2x + y = 4

Add the two equations:
(x - y) + (2x + y) = -1 + 4 → 3x = 3 → x = 1

Plug into first: 1 - y = -1 → -y = -2 → y = 2

Table: x=1, y=2 → L: x=1, y=2 → YES! L

Problem 6 → L

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Problem 7:
2x - y = 14
5x + y = 14

Add them:
(2x - y) + (5x + y) = 14 + 14 → 7x = 28 → x = 4

Plug into first: 2(4) - y = 14 → 8 - y = 14 → -y = 6 → y = -6

Table: x=4, y=-6 → T: x=4, y=-6 → YES! T

Problem 7 → T

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Problem 8:
x + y = -7
x - y = -3

Add them:
(x + y) + (x - y) = -7 + (-3) → 2x = -10 → x = -5

Plug into first: -5 + y = -7 → y = -2

Table: x=-5, y=-2 → P: x=-5, y=-2 → YES! P

Problem 8 → P

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Problem 9:
2x + 3y = 1
5x + 4y = -1

Use elimination. Multiply first by 4, second by 3:

First ×4: 8x + 12y = 4
Second ×3: 15x + 12y = -3

Subtract first from second:
(15x + 12y) - (8x + 12y) = -3 - 4 → 7x = -7 → x = -1

Plug into first: 2(-1) + 3y = 1 → -2 + 3y = 1 → 3y = 3 → y = 1

Table: x=-1, y=1 → F: x=-1, y=1 → YES! F

Problem 9 → F

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Problem 10:
3x - 9 = -4y → rewrite as: 3x + 4y = 9
2x - y = -16 → multiply by 4: 8x - 4y = -64

Now add to first equation:
(3x + 4y) + (8x - 4y) = 9 + (-64) → 11x = -55 → x = -5

Plug into second original: 2(-5) - y = -16 → -10 - y = -16 → -y = -6 → y = 6

Table: x=-5, y=6 → V: x=-5, y=6 → YES! V

Problem 10 → V

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Problem 11:
6x - 5y = -32
5x + 4y = 6

Eliminate y. Multiply first by 4, second by 5:

First ×4: 24x - 20y = -128
Second ×5: 25x + 20y = 30

Add them:
(24x - 20y) + (25x + 20y) = -128 + 30 → 49x = -98 → x = -2

Plug into second: 5(-2) + 4y = 6 → -10 + 4y = 6 → 4y = 16 → y = 4

Table: x=-2, y=4 → A: x=-2, y=4 → YES! A

Problem 11 → A

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Problem 12:
6x - 3y + 27 = 0 → rewrite: 6x - 3y = -27
5x + 4y = 10

Simplify first equation: divide by 3 → 2x - y = -9 → so y = 2x + 9

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

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

Table: x=-2, y=5 → R: x=-2, y=5 → YES! R

Problem 12 → R

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Now, let’s collect all answers:

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

The bottom has four boxes:

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

Final Answer:
SDK BZL TPF VAR