Simultaneous Equations Examples :: A Transum Worksheet - Free Printable

To solve these simultaneous equations, we’ll use the elimination method — that means we’ll add or subtract the equations to get rid of one variable, then solve for the other.

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Let’s go one by one.

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Problem 1:
> 3a + 2b = 12
> 2a + 2b = 10

Subtract the second equation from the first:

(3a + 2b) - (2a + 2b) = 12 - 10
→ 3a - 2a + 2b - 2b = 2
→ a = 2

Now plug a = 2 into one of the equations, say the second:

2(2) + 2b = 10
→ 4 + 2b = 10
→ 2b = 6
→ b = 3

✔ Check: First equation: 3(2) + 2(3) = 6 + 6 = 12 ✔️
Second: 2(2) + 2(3) = 4 + 6 = 10 ✔️

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Problem 2:
> -5x + 3y = 3
> 7x - 3y = 3

Add the two equations together — notice +3y and -3y cancel out!

(-5x + 3y) + (7x - 3y) = 3 + 3
→ 2x = 6
→ x = 3

Plug x = 3 into first equation:

-5(3) + 3y = 3
→ -15 + 3y = 3
→ 3y = 18
→ y = 6

✔ Check: Second equation: 7(3) - 3(6) = 21 - 18 = 3 ✔️

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Problem 3:
> 2c + 2d = 24
> c + 3d = 24

Simplify first equation by dividing by 2:

→ c + d = 12 → Equation A
Second equation: c + 3d = 24 → Equation B

Subtract Equation A from B:

(c + 3d) - (c + d) = 24 - 12
→ 2d = 12
→ d = 6

Plug into Equation A: c + 6 = 12 → c = 6

✔ Check: First original: 2(6)+2(6)=12+12=24 ✔️
Second: 6 + 3(6)=6+18=24 ✔️

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Problem 4:
> 12x + 3y = 60
> 6x - 6y = 0

Simplify second equation: divide by 6 → x - y = 0 → x = y

Plug x = y into first equation:

12x + 3x = 60 → 15x = 60 → x = 4
Then y = 4

✔ Check: First: 12(4)+3(4)=48+12=60 ✔️
Second: 6(4)-6(4)=24-24=0 ✔️

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Problem 5:
> 4e + 6f = 46
> 5e + 4f = 47

Let’s eliminate one variable. Multiply first equation by 2, second by 3 so f coefficients become 12:

First ×2: 8e + 12f = 92
Second ×3: 15e + 12f = 141

Now subtract first from second:

(15e + 12f) - (8e + 12f) = 141 - 92
→ 7e = 49 → e = 7

Plug e = 7 into first original equation:

4(7) + 6f = 46 → 28 + 6f = 46 → 6f = 18 → f = 3

✔ Check: Second equation: 5(7)+4(3)=35+12=47 ✔️

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Problem 6:
> -9x + 5y = 5
> -10x + 3y = -20

Let’s eliminate y. Multiply first by 3, second by 5:

First ×3: -27x + 15y = 15
Second ×5: -50x + 15y = -100

Subtract first from second:

(-50x + 15y) - (-27x + 15y) = -100 - 15
→ -50x + 27x = -115
→ -23x = -115 → x = 5

Plug x = 5 into first equation:

-9(5) + 5y = 5 → -45 + 5y = 5 → 5y = 50 → y = 10

✔ Check: Second equation: -10(5)+3(10)= -50+30= -20 ✔️

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Problem 7:
> g - h = 8
> 2h = g - 5

Rewrite second equation: g = 2h + 5

Plug into first equation:

(2h + 5) - h = 8
→ h + 5 = 8 → h = 3

Then g = 2(3) + 5 = 6 + 5 = 11

✔ Check: First: 11 - 3 = 8 ✔️
Second: 2(3) = 6, and 11 - 5 = 6 ✔️

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Final Answers:

1. a = 2, b = 3
2. x = 3, y = 6
3. c = 6, d = 6
4. x = 4, y = 4
5. e = 7, f = 3
6. x = 5, y = 10
7. g = 11, h = 3

Final Answer:
a=2, b=3; x=3, y=6; c=6, d=6; x=4, y=4; e=7, f=3; x=5, y=10; g=11, h=3