Combining Like Terms and Solving Simple Linear Equations (A) - Free Printable

Problem: Solve each equation to determine the value of the unknown variable.

#### Equation 1: \( 2(3 - h) - 6 = -5h \)

1. Distribute the 2:
\[
2(3 - h) - 6 = 6 - 2h - 6
\]
2. Simplify:
\[
6 - 2h - 6 = -2h
\]
3. The equation becomes:
\[
-2h = -5h
\]
4. Add \(5h\) to both sides:
\[
-2h + 5h = 0
\]
\[
3h = 0
\]
5. Divide by 3:
\[
h = 0
\]

Solution: \( h = 0 \)

---

#### Equation 2: \( 7 + 9d = 7d + 3 \)

1. Subtract \(7d\) from both sides:
\[
7 + 9d - 7d = 7d + 3 - 7d
\]
\[
7 + 2d = 3
\]
2. Subtract 7 from both sides:
\[
7 + 2d - 7 = 3 - 7
\]
\[
2d = -4
\]
3. Divide by 2:
\[
d = -2
\]

Solution: \( d = -2 \)

---

#### Equation 3: \( -2(4 + 3y) = -2(4 + y) \)

1. Distribute the \(-2\) on both sides:
\[
-2(4 + 3y) = -8 - 6y
\]
\[
-2(4 + y) = -8 - 2y
\]
2. The equation becomes:
\[
-8 - 6y = -8 - 2y
\]
3. Add 8 to both sides:
\[
-8 - 6y + 8 = -8 - 2y + 8
\]
\[
-6y = -2y
\]
4. Add \(6y\) to both sides:
\[
-6y + 6y = -2y + 6y
\]
\[
0 = 4y
\]
5. Divide by 4:
\[
y = 0
\]

Solution: \( y = 0 \)

---

#### Equation 4: \( -7 + 4c = 7c + 6 \)

1. Subtract \(4c\) from both sides:
\[
-7 + 4c - 4c = 7c + 6 - 4c
\]
\[
-7 = 3c + 6
\]
2. Subtract 6 from both sides:
\[
-7 - 6 = 3c + 6 - 6
\]
\[
-13 = 3c
\]
3. Divide by 3:
\[
c = -\frac{13}{3}
\]

Solution: \( c = -\frac{13}{3} \)

---

#### Equation 5: \( 5(1 + s) = -9s + 6 \)

1. Distribute the 5:
\[
5(1 + s) = 5 + 5s
\]
2. The equation becomes:
\[
5 + 5s = -9s + 6
\]
3. Add \(9s\) to both sides:
\[
5 + 5s + 9s = -9s + 6 + 9s
\]
\[
5 + 14s = 6
\]
4. Subtract 5 from both sides:
\[
5 + 14s - 5 = 6 - 5
\]
\[
14s = 1
\]
5. Divide by 14:
\[
s = \frac{1}{14}
\]

Solution: \( s = \frac{1}{14} \)

---

#### Equation 6: \( 3 + v = 2(2v - 1) \)

1. Distribute the 2:
\[
2(2v - 1) = 4v - 2
\]
2. The equation becomes:
\[
3 + v = 4v - 2
\]
3. Subtract \(v\) from both sides:
\[
3 + v - v = 4v - 2 - v
\]
\[
3 = 3v - 2
\]
4. Add 2 to both sides:
\[
3 + 2 = 3v - 2 + 2
\]
\[
5 = 3v
\]
5. Divide by 3:
\[
v = \frac{5}{3}
\]

Solution: \( v = \frac{5}{3} \)

---

#### Equation 7: \( -2 - 4w = 7w - 8 \)

1. Add \(4w\) to both sides:
\[
-2 - 4w + 4w = 7w - 8 + 4w
\]
\[
-2 = 11w - 8
\]
2. Add 8 to both sides:
\[
-2 + 8 = 11w - 8 + 8
\]
\[
6 = 11w
\]
3. Divide by 11:
\[
w = \frac{6}{11}
\]

Solution: \( w = \frac{6}{11} \)

---

#### Equation 8: \( -6(1 - m) = 9 - 2m \)

1. Distribute the \(-6\):
\[
-6(1 - m) = -6 + 6m
\]
2. The equation becomes:
\[
-6 + 6m = 9 - 2m
\]
3. Add \(2m\) to both sides:
\[
-6 + 6m + 2m = 9 - 2m + 2m
\]
\[
-6 + 8m = 9
\]
4. Add 6 to both sides:
\[
-6 + 8m + 6 = 9 + 6
\]
\[
8m = 15
\]
5. Divide by 8:
\[
m = \frac{15}{8}
\]

Solution: \( m = \frac{15}{8} \)

---

#### Equation 9: \( -2q - 3 = -2(2q + 1) \)

1. Distribute the \(-2\):
\[
-2(2q + 1) = -4q - 2
\]
2. The equation becomes:
\[
-2q - 3 = -4q - 2
\]
3. Add \(4q\) to both sides:
\[
-2q - 3 + 4q = -4q - 2 + 4q
\]
\[
2q - 3 = -2
\]
4. Add 3 to both sides:
\[
2q - 3 + 3 = -2 + 3
\]
\[
2q = 1
\]
5. Divide by 2:
\[
q = \frac{1}{2}
\]

Solution: \( q = \frac{1}{2} \)

---

#### Equation 10: \( 6n + 7 = 2n + 5 \)

1. Subtract \(2n\) from both sides:
\[
6n + 7 - 2n = 2n + 5 - 2n
\]
\[
4n + 7 = 5
\]
2. Subtract 7 from both sides:
\[
4n + 7 - 7 = 5 - 7
\]
\[
4n = -2
\]
3. Divide by 4:
\[
n = -\frac{1}{2}
\]

Solution: \( n = -\frac{1}{2} \)

---

#### Equation 11: \( 2(3x - 2) + 9 = -5x \)

1. Distribute the 2:
\[
2(3x - 2) = 6x - 4
\]
2. The equation becomes:
\[
6x - 4 + 9 = -5x
\]
3. Simplify:
\[
6x + 5 = -5x
\]
4. Add \(5x\) to both sides:
\[
6x + 5 + 5x = -5x + 5x
\]
\[
11x + 5 = 0
\]
5. Subtract 5 from both sides:
\[
11x + 5 - 5 = 0 - 5
\]
\[
11x = -5
\]
6. Divide by 11:
\[
x = -\frac{5}{11}
\]

Solution: \( x = -\frac{5}{11} \)

---

#### Equation 12: \( 3(1 + p) = -5(p + 1) \)

1. Distribute on both sides:
\[
3(1 + p) = 3 + 3p
\]
\[
-5(p + 1) = -5p - 5
\]
2. The equation becomes:
\[
3 + 3p = -5p - 5
\]
3. Add \(5p\) to both sides:
\[
3 + 3p + 5p = -5p - 5 + 5p
\]
\[
3 + 8p = -5
\]
4. Subtract 3 from both sides:
\[
3 + 8p - 3 = -5 - 3
\]
\[
8p = -8
\]
5. Divide by 8:
\[
p = -1
\]

Solution: \( p = -1 \)

---

#### Equation 13: \( 3(1 - 3g) = -7 + g \)

1. Distribute the 3:
\[
3(1 - 3g) = 3 - 9g
\]
2. The equation becomes:
\[
3 - 9g = -7 + g
\]
3. Subtract \(g\) from both sides:
\[
3 - 9g - g = -7 + g - g
\]
\[
3 - 10g = -7
\]
4. Subtract 3 from both sides:
\[
3 - 10g - 3 = -7 - 3
\]
\[
-10g = -10
\]
5. Divide by \(-10\):
\[
g = 1
\]

Solution: \( g = 1 \)

---

#### Equation 14: \( 1 + 2b = 4b + 9 \)

1. Subtract \(2b\) from both sides:
\[
1 + 2b - 2b = 4b + 9 - 2b
\]
\[
1 = 2b + 9
\]
2. Subtract 9 from both sides:
\[
1 - 9 = 2b + 9 - 9
\]
\[
-8 = 2b
\]
3. Divide by 2:
\[
b = -4
\]

Solution: \( b = -4 \)

---

#### Equation 15: \( 2z + 6 = 3z + 1 \)

1. Subtract \(2z\) from both sides:
\[
2z + 6 - 2z = 3z + 1 - 2z
\]
\[
6 = z + 1
\]
2. Subtract 1 from both sides:
\[
6 - 1 = z + 1 - 1
\]
\[
z = 5
\]

Solution: \( z = 5 \)

---

#### Equation 16: \( 5a - 2 = -9a + 8 \)

1. Add \(9a\) to both sides:
\[
5a - 2 + 9a = -9a + 8 + 9a
\]
\[
14a - 2 = 8
\]
2. Add 2 to both sides:
\[
14a - 2 + 2 = 8 + 2
\]
\[
14a = 10
\]
3. Divide by 14:
\[
a = \frac{10}{14} = \frac{5}{7}
\]

Solution: \( a = \frac{5}{7} \)

---

#### Equation 17: \( 6t - 5 = -9t - 9 \)

1. Add \(9t\) to both sides:
\[
6t - 5 + 9t = -9t - 9 + 9t
\]
\[
15t - 5 = -9
\]
2. Add 5 to both sides:
\[
15t - 5 + 5 = -9 + 5
\]
\[
15t = -4
\]
3. Divide by 15:
\[
t = -\frac{4}{15}
\]

Solution: \( t = -\frac{4}{15} \)

---

#### Equation 18: \( -1 + 3f = -7 - 6f \)

1. Add \(6f\) to both sides:
\[
-1 + 3f + 6f = -7 - 6f + 6f
\]
\[
-1 + 9f = -7
\]
2. Add 1 to both sides:
\[
-1 + 9f + 1 = -7 + 1
\]
\[
9f = -6
\]
3. Divide by 9:
\[
f = -\frac{6}{9} = -\frac{2}{3}
\]

Solution: \( f = -\frac{2}{3} \)

---

#### Equation 19: \( 2 + r = 7 + 6r \)

1. Subtract \(r\) from both sides:
\[
2 + r - r = 7 + 6r - r
\]
\[
2 = 7 + 5r
\]
2. Subtract 7 from both sides:
\[
2 - 7 = 7 + 5r - 7
\]
\[
-5 = 5r
\]
3. Divide by 5:
\[
r = -1
\]

Solution: \( r = -1 \)

---

#### Equation 20: \( -6k + 1 = -2 + 7k \)

1. Add \(6k\) to both sides:
\[
-6k + 1 + 6k = -2 + 7k + 6k
\]
\[
1 = -2 + 13k
\]
2. Add 2 to both sides:
\[
1 + 2 = -2 + 13k + 2
\]
\[
3 = 13k
\]
3. Divide by 13:
\[
k = \frac{3}{13}
\]

Solution: \( k = \frac{3}{13} \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ h = 0 \\
2. & \ d = -2 \\
3. & \ y = 0 \\
4. & \ c = -\frac{13}{3} \\
5. & \ s = \frac{1}{14} \\
6. & \ v = \frac{5}{3} \\
7. & \ w = \frac{6}{11} \\
8. & \ m = \frac{15}{8} \\
9. & \ q = \frac{1}{2} \\
10. & \ n = -\frac{1}{2} \\
11. & \ x = -\frac{5}{11} \\
12. & \ p = -1 \\
13. & \ g = 1 \\
14. & \ b = -4 \\
15. & \ z = 5 \\
16. & \ a = \frac{5}{7} \\
17. & \ t = -\frac{4}{15} \\
18. & \ f = -\frac{2}{3} \\
19. & \ r = -1 \\
20. & \ k = \frac{3}{13}
\end{aligned}
}
\]