Linear Systems (A) worksheet featuring eight systems of equations to solve.

Let’s solve each system of equations step by step. We’ll use substitution or elimination — whichever is easier for each pair.

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Problem 1:
3u + z = 15
u + 2z = 10

Solve the second equation for u:
u = 10 - 2z

Plug into first equation:
3(10 - 2z) + z = 15
30 - 6z + z = 15
30 - 5z = 15
-5z = -15
z = 3

Now plug back to find u:
u = 10 - 2(3) = 10 - 6 = 4

✔ Solution: u = 4, z = 3

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Problem 2:
u + 6y = 32
u + 3y = 17

Subtract second equation from first:
(u + 6y) - (u + 3y) = 32 - 17
3y = 15 → y = 5

Plug into second equation:
u + 3(5) = 17 → u + 15 = 17 → u = 2

✔ Solution: u = 2, y = 5

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Problem 3:
3c + 4u = 33
6c + 3u = 36

Multiply first equation by 2:
6c + 8u = 66

Now subtract second equation:
(6c + 8u) - (6c + 3u) = 66 - 36
5u = 30 → u = 6

Plug into first equation:
3c + 4(6) = 33 → 3c + 24 = 33 → 3c = 9 → c = 3

✔ Solution: c = 3, u = 6

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Problem 4:
6u + v = 18
5u + 2v = 22

Solve first equation for v:
v = 18 - 6u

Plug into second:
5u + 2(18 - 6u) = 22
5u + 36 - 12u = 22
-7u + 36 = 22
-7u = -14 → u = 2

Then v = 18 - 6(2) = 18 - 12 = 6

✔ Solution: u = 2, v = 6

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Problem 5:
2a + 2x = 18
a + 3x = 17

Simplify first equation: divide by 2 → a + x = 9 → a = 9 - x

Plug into second:
(9 - x) + 3x = 17
9 + 2x = 17
2x = 8 → x = 4

Then a = 9 - 4 = 5

✔ Solution: a = 5, x = 4

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Problem 6:
5a + 2v = 32
6a + 6v = 42

Simplify second equation: divide by 6 → a + v = 7 → v = 7 - a

Plug into first:
5a + 2(7 - a) = 32
5a + 14 - 2a = 32
3a + 14 = 32
3a = 18 → a = 6

Then v = 7 - 6 = 1

✔ Solution: a = 6, v = 1

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Problem 7:
2b + v = 13
b + v = 8

Subtract second from first:
(2b + v) - (b + v) = 13 - 8
b = 5

Plug into second: 5 + v = 8 → v = 3

✔ Solution: b = 5, v = 3

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Problem 8:
3a + 5u = 17
2a + u = 9

Solve second for u: u = 9 - 2a

Plug into first:
3a + 5(9 - 2a) = 17
3a + 45 - 10a = 17
-7a + 45 = 17
-7a = -28 → a = 4

Then u = 9 - 2(4) = 9 - 8 = 1

✔ Solution: a = 4, u = 1

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Final Answer:

1. u = 4, z = 3
2. u = 2, y = 5
3. c = 3, u = 6
4. u = 2, v = 6
5. a = 5, x = 4
6. a = 6, v = 1
7. b = 5, v = 3
8. a = 4, u = 1