Solving Equations With Brackets - ANSWERS | PDF - Free Printable

Problem: Solving Equations with Brackets

The task involves solving a series of equations that involve brackets. We will solve each equation step by step, explaining the process clearly.

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#### Question 1 – Complete these to test the waters!

##### a) \( 2(y + 2) = 8 \)

1. Distribute the 2 across the brackets:
\[
2(y + 2) = 2y + 4
\]
2. Set the equation equal to 8:
\[
2y + 4 = 8
\]
3. Subtract 4 from both sides:
\[
2y = 4
\]
4. Divide by 2:
\[
y = 2
\]

Answer: \( y = 2 \)

##### b) \( 3(a + 2) = 12 \)

1. Distribute the 3 across the brackets:
\[
3(a + 2) = 3a + 6
\]
2. Set the equation equal to 12:
\[
3a + 6 = 12
\]
3. Subtract 6 from both sides:
\[
3a = 6
\]
4. Divide by 3:
\[
a = 2
\]

Answer: \( a = 2 \)

##### c) \( 4(x - 4) = 40 \)

1. Distribute the 4 across the brackets:
\[
4(x - 4) = 4x - 16
\]
2. Set the equation equal to 40:
\[
4x - 16 = 40
\]
3. Add 16 to both sides:
\[
4x = 56
\]
4. Divide by 4:
\[
x = 14
\]

Answer: \( x = 14 \)

##### d) \( 5(c - 3) = 35 \)

1. Distribute the 5 across the brackets:
\[
5(c - 3) = 5c - 15
\]
2. Set the equation equal to 35:
\[
5c - 15 = 35
\]
3. Add 15 to both sides:
\[
5c = 50
\]
4. Divide by 5:
\[
c = 10
\]

Answer: \( c = 10 \)

##### e) \( 7(3 + a) = 49 \)

1. Distribute the 7 across the brackets:
\[
7(3 + a) = 21 + 7a
\]
2. Set the equation equal to 49:
\[
21 + 7a = 49
\]
3. Subtract 21 from both sides:
\[
7a = 28
\]
4. Divide by 7:
\[
a = 4
\]

Answer: \( a = 4 \)

##### f) \( (-2 + x) = 36 \)

1. Simplify the left side (no distribution needed):
\[
-2 + x = 36
\]
2. Add 2 to both sides:
\[
x = 38
\]

Answer: \( x = 38 \)

##### g) \( 6(6 + p) = 48 \)

1. Distribute the 6 across the brackets:
\[
6(6 + p) = 36 + 6p
\]
2. Set the equation equal to 48:
\[
36 + 6p = 48
\]
3. Subtract 36 from both sides:
\[
6p = 12
\]
4. Divide by 6:
\[
p = 2
\]

Answer: \( p = 2 \)

##### h) \( 9(x + 4) = 18 \)

1. Distribute the 9 across the brackets:
\[
9(x + 4) = 9x + 36
\]
2. Set the equation equal to 18:
\[
9x + 36 = 18
\]
3. Subtract 36 from both sides:
\[
9x = -18
\]
4. Divide by 9:
\[
x = -2
\]

Answer: \( x = -2 \)

##### i) \( 4(a + 6) = 4 \)

1. Distribute the 4 across the brackets:
\[
4(a + 6) = 4a + 24
\]
2. Set the equation equal to 4:
\[
4a + 24 = 4
\]
3. Subtract 24 from both sides:
\[
4a = -20
\]
4. Divide by 4:
\[
a = -5
\]

Answer: \( a = -5 \)

##### j) \( 8(3 + m) = 8 \)

1. Distribute the 8 across the brackets:
\[
8(3 + m) = 24 + 8m
\]
2. Set the equation equal to 8:
\[
24 + 8m = 8
\]
3. Subtract 24 from both sides:
\[
8m = -16
\]
4. Divide by 8:
\[
m = -2
\]

Answer: \( m = -2 \)

---

#### Question 2 – More practice? Work here

##### a) \( 2(4x + 8) = 32 \)

1. Distribute the 2 across the brackets:
\[
2(4x + 8) = 8x + 16
\]
2. Set the equation equal to 32:
\[
8x + 16 = 32
\]
3. Subtract 16 from both sides:
\[
8x = 16
\]
4. Divide by 8:
\[
x = 2
\]

Answer: \( x = 2 \)

##### b) \( 5(7x - 4) = 15 \)

1. Distribute the 5 across the brackets:
\[
5(7x - 4) = 35x - 20
\]
2. Set the equation equal to 15:
\[
35x - 20 = 15
\]
3. Add 20 to both sides:
\[
35x = 35
\]
4. Divide by 35:
\[
x = 1
\]

Answer: \( x = 1 \)

##### c) \( 5(5x - 3) = 35 \)

1. Distribute the 5 across the brackets:
\[
5(5x - 3) = 25x - 15
\]
2. Set the equation equal to 35:
\[
25x - 15 = 35
\]
3. Add 15 to both sides:
\[
25x = 50
\]
4. Divide by 25:
\[
x = 2
\]

Answer: \( x = 2 \)

##### d) \( 4(6x + 3) = 36 \)

1. Distribute the 4 across the brackets:
\[
4(6x + 3) = 24x + 12
\]
2. Set the equation equal to 36:
\[
24x + 12 = 36
\]
3. Subtract 12 from both sides:
\[
24x = 24
\]
4. Divide by 24:
\[
x = 1
\]

Answer: \( x = 1 \)

##### e) \( 7(2x - 7) = 7 \)

1. Distribute the 7 across the brackets:
\[
7(2x - 7) = 14x - 49
\]
2. Set the equation equal to 7:
\[
14x - 49 = 7
\]
3. Add 49 to both sides:
\[
14x = 56
\]
4. Divide by 14:
\[
x = 4
\]

Answer: \( x = 4 \)

##### f) \( 6(2x - 3) = 42 \)

1. Distribute the 6 across the brackets:
\[
6(2x - 3) = 12x - 18
\]
2. Set the equation equal to 42:
\[
12x - 18 = 42
\]
3. Add 18 to both sides:
\[
12x = 60
\]
4. Divide by 12:
\[
x = 5
\]

Answer: \( x = 5 \)

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#### Question 3 – Need a bigger challenge? Go!

##### a) \( 3(2x + 1) + 2(4x + 2) = 35 \)

1. Distribute in both terms:
\[
3(2x + 1) = 6x + 3
\]
\[
2(4x + 2) = 8x + 4
\]
2. Combine the expressions:
\[
6x + 3 + 8x + 4 = 35
\]
3. Simplify:
\[
14x + 7 = 35
\]
4. Subtract 7 from both sides:
\[
14x = 28
\]
5. Divide by 14:
\[
x = 2
\]

Answer: \( x = 2 \)

##### b) \( 2(x + 3) + 3(x + 1) = 24 \)

1. Distribute in both terms:
\[
2(x + 3) = 2x + 6
\]
\[
3(x + 1) = 3x + 3
\]
2. Combine the expressions:
\[
2x + 6 + 3x + 3 = 24
\]
3. Simplify:
\[
5x + 9 = 24
\]
4. Subtract 9 from both sides:
\[
5x = 15
\]
5. Divide by 5:
\[
x = 3
\]

Answer: \( x = 3 \)

##### c) \( 4(3x - 2) + 8(x + 1) = 100 \)

1. Distribute in both terms:
\[
4(3x - 2) = 12x - 8
\]
\[
8(x + 1) = 8x + 8
\]
2. Combine the expressions:
\[
12x - 8 + 8x + 8 = 100
\]
3. Simplify:
\[
20x = 100
\]
4. Divide by 20:
\[
x = 5
\]

Answer: \( x = 5 \)

##### d) \( 6(x + 2) + 4(3 - x) = 30 \)

1. Distribute in both terms:
\[
6(x + 2) = 6x + 12
\]
\[
4(3 - x) = 12 - 4x
\]
2. Combine the expressions:
\[
6x + 12 + 12 - 4x = 30
\]
3. Simplify:
\[
2x + 24 = 30
\]
4. Subtract 24 from both sides:
\[
2x = 6
\]
5. Divide by 2:
\[
x = 3
\]

Answer: \( x = 3 \)

##### e) \( 5(2x + 3) + 2(5x + 1) = 37 \)

1. Distribute in both terms:
\[
5(2x + 3) = 10x + 15
\]
\[
2(5x + 1) = 10x + 2
\]
2. Combine the expressions:
\[
10x + 15 + 10x + 2 = 37
\]
3. Simplify:
\[
20x + 17 = 37
\]
4. Subtract 17 from both sides:
\[
20x = 20
\]
5. Divide by 20:
\[
x = 1
\]

Answer: \( x = 1 \)

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#### Question 4 – The Big Time – These are tough!

##### a) \( 2(3x + 1) - 3(x + 2) = 2 \)

1. Distribute in both terms:
\[
2(3x + 1) = 6x + 2
\]
\[
3(x + 2) = 3x + 6
\]
2. Substitute back into the equation:
\[
6x + 2 - (3x + 6) = 2
\]
3. Simplify the subtraction:
\[
6x + 2 - 3x - 6 = 2
\]
4. Combine like terms:
\[
3x - 4 = 2
\]
5. Add 4 to both sides:
\[
3x = 6
\]
6. Divide by 3:
\[
x = 2
\]

Answer: \( x = 2 \)

##### b) \( 5(2x + 3) - 3(3x - 2) = 22 \)

1. Distribute in both terms:
\[
5(2x + 3) = 10x + 15
\]
\[
3(3x - 2) = 9x - 6
\]
2. Substitute back into the equation:
\[
10x + 15 - (9x - 6) = 22
\]
3. Simplify the subtraction:
\[
10x + 15 - 9x + 6 = 22
\]
4. Combine like terms:
\[
x + 21 = 22
\]
5. Subtract 21 from both sides:
\[
x = 1
\]

Answer: \( x = 1 \)

##### c) \( 4(2x - 3) + 2(x - 4) = 10 \)

1. Distribute in both terms:
\[
4(2x - 3) = 8x - 12
\]
\[
2(x - 4) = 2x - 8
\]
2. Substitute back into the equation:
\[
8x - 12 + 2x - 8 = 10
\]
3. Combine like terms:
\[
10x - 20 = 10
\]
4. Add 20 to both sides:
\[
10x = 30
\]
5. Divide by 10:
\[
x = 3
\]

Answer: \( x = 3 \)

##### d) \( 3(4x - 2) + 5(x - 1) = 23 \)

1. Distribute in both terms:
\[
3(4x - 2) = 12x - 6
\]
\[
5(x - 1) = 5x - 5
\]
2. Substitute back into the equation:
\[
12x - 6 + 5x - 5 = 23
\]
3. Combine like terms:
\[
17x - 11 = 23
\]
4. Add 11 to both sides:
\[
17x = 34
\]
5. Divide by 17:
\[
x = 2
\]

Answer: \( x = 2 \)

##### e) \( 8(2x - 1) - 3(5x - 3) = 1 \)

1. Distribute in both terms:
\[
8(2x - 1) = 16x - 8
\]
\[
3(5x - 3) = 15x - 9
\]
2. Substitute back into the equation:
\[
16x - 8 - (15x - 9) = 1
\]
3. Simplify the subtraction:
\[
16x - 8 - 15x + 9 = 1
\]
4. Combine like terms:
\[
x + 1 = 1
\]
5. Subtract 1 from both sides:
\[
x = 0
\]

Answer: \( x = 0 \)

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Final Answers:

1. \( y = 2 \)
2. \( a = 2 \)
3. \( x = 14 \)
4. \( c = 10 \)
5. \( a = 4 \)
6. \( x = 38 \)
7. \( p = 2 \)
8. \( x = -2 \)
9. \( a = -5 \)
10. \( m = -2 \)
11. \( x = 2 \)
12. \( x = 1 \)
13. \( x = 2 \)
14. \( x = 1 \)
15. \( x = 4 \)
16. \( x = 5 \)
17. \( x = 2 \)
18. \( x = 3 \)
19. \( x = 5 \)
20. \( x = 3 \)
21. \( x = 1 \)
22. \( x = 2 \)
23. \( x = 1 \)
24. \( x = 3 \)
25. \( x = 2 \)
26. \( x = 0 \)

\boxed{x = 0}