Simultaneous equations questions - Top KS3/4 worksheets & lessons ... - Free Printable

We are given a worksheet on Simultaneous Equations. Since you mentioned “I uploaded an image with a task,” and the image shows 16 problems (A1 to D4), I’ll solve all 16 problems step-by-step, explaining the method used for each.

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## ✔ GENERAL METHODS FOR SIMULTANEOUS EQUATIONS:

There are two main methods:
1. Elimination Method: Add or subtract equations to eliminate one variable.
2. Substitution Method: Solve one equation for one variable and substitute into the other.

I’ll use whichever is most efficient for each problem.

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# 🔹 SECTION A

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A1:

> x + y = 1
> x - y = 5

✔ Method: Elimination

Add both equations:
(x + y) + (x - y) = 1 + 5
→ 2x = 6 → x = 3

Substitute x = 3 into first equation:
3 + y = 1 → y = -2

✔ Answer: x = 3, y = -2

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A2:

> 5x + y = 17
> x + y = 3

✔ Method: Elimination

Subtract second from first:
(5x + y) - (x + y) = 17 - 3
→ 4x = 14 → x = 3.5

Substitute into x + y = 3:
3.5 + y = 3 → y = -0.5

✔ Answer: x = 3.5, y = -0.5

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A3:

> 6x - 5y = 9
> 6x + 3y = 33

✔ Method: Elimination

Subtract first from second:
(6x + 3y) - (6x - 5y) = 33 - 9
→ 8y = 24 → y = 3

Substitute y = 3 into first equation:
6x - 5(3) = 9 → 6x - 15 = 9 → 6x = 24 → x = 4

✔ Answer: x = 4, y = 3

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A4:

> x + 5y = -13
> 4x - 5y = 48

✔ Method: Elimination

Add both equations:
(x + 5y) + (4x - 5y) = -13 + 48
→ 5x = 35 → x = 7

Substitute into first equation:
7 + 5y = -13 → 5y = -20 → y = -4

✔ Answer: x = 7, y = -4

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# 🔹 SECTION B

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B1:

> 8x - y = 7
> 12x - 8y = 6

✔ Method: Substitution

From first equation:
y = 8x - 7

Substitute into second:
12x - 8(8x - 7) = 6
→ 12x - 64x + 56 = 6
→ -52x = -50 → x = 50/52 = 25/26

Then y = 8*(25/26) - 7 = 200/26 - 182/26 = 18/26 = 9/13

✔ Answer: x = 25/26, y = 9/13

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B2:

> 3x - 2y = 13
> x - y = 5

✔ Method: Substitution

From second: x = y + 5

Substitute into first:
3(y + 5) - 2y = 13
→ 3y + 15 - 2y = 13
→ y + 15 = 13 → y = -2

Then x = -2 + 5 = 3

✔ Answer: x = 3, y = -2

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B3:

> 2x - 3y = 3
> 3x + 6y = 1

✔ Method: Elimination

Multiply first by 2:
4x - 6y = 6

Now add to second:
(4x - 6y) + (3x + 6y) = 6 + 1
→ 7x = 7 → x = 1

Substitute into first:
2(1) - 3y = 3 → 2 - 3y = 3 → -3y = 1 → y = -1/3

✔ Answer: x = 1, y = -1/3

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B4:

> 3x + 5y = 7
> 9x + 11y = 13

✔ Method: Elimination

Multiply first by 3:
9x + 15y = 21

Subtract second:
(9x + 15y) - (9x + 11y) = 21 - 13
→ 4y = 8 → y = 2

Substitute into first:
3x + 5(2) = 7 → 3x + 10 = 7 → 3x = -3 → x = -1

✔ Answer: x = -1, y = 2

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# 🔹 SECTION C

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C1:

> 5x + 3y = 14
> 2x + 2y = 4

Simplify second equation: divide by 2 → x + y = 2

So now:
5x + 3y = 14
x + y = 2

✔ Method: Substitution

From second: x = 2 - y

Substitute:
5(2 - y) + 3y = 14
→ 10 - 5y + 3y = 14
→ -2y = 4 → y = -2

Then x = 2 - (-2) = 4

✔ Answer: x = 4, y = -2

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C2:

> 4x + 5y = 13
> 3x - 2y = 27

✔ Method: Elimination

Multiply first by 2, second by 5:
8x + 10y = 26
15x - 10y = 135

Add:
23x = 161 → x = 7

Substitute into first:
4(7) + 5y = 13 → 28 + 5y = 13 → 5y = -15 → y = -3

✔ Answer: x = 7, y = -3

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C3:

> 2x + 7y = 12
> 3x + 8y = 13

✔ Method: Elimination

Multiply first by 3, second by 2:
6x + 21y = 36
6x + 16y = 26

Subtract:
(6x + 21y) - (6x + 16y) = 36 - 26
→ 5y = 10 → y = 2

Substitute into first:
2x + 7(2) = 12 → 2x + 14 = 12 → 2x = -2 → x = -1

✔ Answer: x = -1, y = 2

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C4:

> 3x + 2y = 7
> 4x - 3y = 15

✔ Method: Elimination

Multiply first by 3, second by 2:
9x + 6y = 21
8x - 6y = 30

Add:
17x = 51 → x = 3

Substitute into first:
3(3) + 2y = 7 → 9 + 2y = 7 → 2y = -2 → y = -1

✔ Answer: x = 3, y = -1

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# 🔹 SECTION D

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D1:

> y = 4x - 8
> y = 3x - 5

✔ Method: Substitution (already solved for y)

Set equal:
4x - 8 = 3x - 5
→ x = 3

Then y = 4(3) - 8 = 12 - 8 = 4

✔ Answer: x = 3, y = 4

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D2:

> y = 4x - 2
> y = 9x - 12

Set equal:
4x - 2 = 9x - 12
→ -5x = -10 → x = 2

Then y = 4(2) - 2 = 8 - 2 = 6

✔ Answer: x = 2, y = 6

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D3:

> 3x + 2y = 61
> y = 2x - 22

✔ Method: Substitution

Substitute y into first:
3x + 2(2x - 22) = 61
→ 3x + 4x - 44 = 61
→ 7x = 105 → x = 15

Then y = 2(15) - 22 = 30 - 22 = 8

✔ Answer: x = 15, y = 8

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D4:

> 3x - 2y = -24
> y = 5 - 2x

✔ Method: Substitution

Substitute y into first:
3x - 2(5 - 2x) = -24
→ 3x - 10 + 4x = -24
→ 7x = -14 → x = -2

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

✔ Answer: x = -2, y = 9

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# 🎯 FINAL ANSWERS SUMMARY:

| Problem | x | y |
|---------|----------|-----------|
| A1 | 3 | -2 |
| A2 | 3.5 | -0.5 |
| A3 | 4 | 3 |
| A4 | 7 | -4 |
| B1 | 25/26 | 9/13 |
| B2 | 3 | -2 |
| B3 | 1 | -1/3 |
| B4 | -1 | 2 |
| C1 | 4 | -2 |
| C2 | 7 | -3 |
| C3 | -1 | 2 |
| C4 | 3 | -1 |
| D1 | 3 | 4 |
| D2 | 2 | 6 |
| D3 | 15 | 8 |
| D4 | -2 | 9 |

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✔ All problems solved using either elimination or substitution — whichever was more straightforward.

Let me know if you’d like to see any specific problem explained in more detail!