Answer :
Let's solve each equation step-by-step.
### Part (a): [tex]\(2(3x + 1) = 8\)[/tex]
1. Distribute the 2 on the left-hand side:
[tex]\[2 \cdot (3x) + 2 \cdot 1 = 8\][/tex]
[tex]\[6x + 2 = 8\][/tex]
2. Subtract 2 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[6x = 8 - 2\][/tex]
[tex]\[6x = 6\][/tex]
3. Divide both sides by 6:
[tex]\[x = \frac{6}{6}\][/tex]
[tex]\[x = 1\][/tex]
So the solution is [tex]\(x = 1\)[/tex].
### Part (b): [tex]\(4 + 18d = [(2 + 2) + 3(2d - 2)]\)[/tex]
1. Simplify inside the brackets on the right-hand side:
[tex]\[(2 + 2) + 3(2d - 2)\][/tex]
[tex]\[4 + 3(2d - 2)\][/tex]
2. Distribute the 3:
[tex]\[4 + 3 \cdot 2d - 3 \cdot 2\][/tex]
[tex]\[4 + 6d - 6\][/tex]
3. Simplify:
[tex]\[4 + 18d = 4 + 6d - 6\][/tex]
[tex]\[4 + 18d = 6d - 2\][/tex]
4. Subtract 6d from both sides:
[tex]\[4 + 12d = -2\][/tex]
5. Subtract 4 from both sides:
[tex]\[12d = -2 - 4\][/tex]
[tex]\[12d = -6\][/tex]
6. Divide by 12:
[tex]\[d = \frac{-6}{12}\][/tex]
[tex]\[d = -\frac{1}{2}\][/tex]
So the solution is [tex]\(d = -\frac{1}{2}\)[/tex].
### Part (c): [tex]\(4(u + 3) = 5u - 3u\)[/tex]
1. Distribute the 4 on the left-hand side:
[tex]\[4u + 12 = 2u\][/tex]
2. Subtract 2u from both sides:
[tex]\[4u - 2u + 12 = 0\][/tex]
[tex]\[2u + 12 = 0\][/tex]
3. Subtract 12 from both sides:
[tex]\[2u = -12\][/tex]
4. Divide by 2:
[tex]\[u = \frac{-12}{2}\][/tex]
[tex]\[u = -6\][/tex]
So the solution is [tex]\(u = -6\)[/tex].
### Part (d): [tex]\(v + 8 = [(3v + 4 - v) - 2] + 4\)[/tex]
1. Simplify inside the brackets on the right-hand side:
[tex]\[ (3v + 4 - v) - 2 + 4\][/tex]
[tex]\[ (2v + 4) - 2 + 4\][/tex]
[tex]\[ 2v + 8 - 2\][/tex]
[tex]\[ 2v + 6\][/tex]
2. Now, the equation is:
[tex]\[v + 8 = 2v + 6\][/tex]
3. Subtract [tex]\(v\)[/tex] from both sides:
[tex]\[8 = v + 6\][/tex]
4. Subtract 6 from both sides:
[tex]\[v = 2\][/tex]
So the solution is [tex]\(v = 2\)[/tex].
### Part (e): [tex]\(\{[(3 + 4) - 6x \cdot 2] + 10\} = 5\)[/tex]
1. Simplify inside the brackets:
[tex]\[(3 + 4) - 6x \cdot 2\][/tex]
[tex]\[7 - 12x\][/tex]
2. Add the 10:
[tex]\[ (7 - 12x) + 10 = 5\][/tex]
[tex]\[ 17 - 12x = 5\][/tex]
3. Subtract 17 from both sides:
[tex]\[-12x = 5 - 17\][/tex]
[tex]\[-12x = -12\][/tex]
4. Divide by -12:
[tex]\[x = \frac{-12}{-12}\][/tex]
[tex]\[x = 1\][/tex]
So the solution is [tex]\(x = 1\)[/tex].
### Part (f): [tex]\(\{[(7 \cdot 2x) - (7 \cdot 7) + 4x] + 85\} = 0\)[/tex]
1. Simplify inside the brackets:
[tex]\[(7 \cdot 2x) - (7 \cdot 7) + 4x\][/tex]
[tex]\[14x - 49 + 4x\][/tex]
[tex]\[18x - 49\][/tex]
2. Add the 85:
[tex]\[ (18x - 49) + 85 = 0\][/tex]
[tex]\[ 18x + 36 = 0\][/tex]
3. Subtract 36 from both sides:
[tex]\[18x = -36\][/tex]
4. Divide by 18:
[tex]\[x = \frac{-36}{18}\][/tex]
[tex]\[x = -2\][/tex]
So the solution is [tex]\(x = -2\)[/tex].
In summary, the solutions are:
a) [tex]\(x = 1\)[/tex]
b) [tex]\(d = -\frac{1}{2}\)[/tex]
c) [tex]\(u = -6\)[/tex]
d) [tex]\(v = 2\)[/tex]
e) [tex]\(x = 1\)[/tex]
f) [tex]\(x = -2\)[/tex]
### Part (a): [tex]\(2(3x + 1) = 8\)[/tex]
1. Distribute the 2 on the left-hand side:
[tex]\[2 \cdot (3x) + 2 \cdot 1 = 8\][/tex]
[tex]\[6x + 2 = 8\][/tex]
2. Subtract 2 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[6x = 8 - 2\][/tex]
[tex]\[6x = 6\][/tex]
3. Divide both sides by 6:
[tex]\[x = \frac{6}{6}\][/tex]
[tex]\[x = 1\][/tex]
So the solution is [tex]\(x = 1\)[/tex].
### Part (b): [tex]\(4 + 18d = [(2 + 2) + 3(2d - 2)]\)[/tex]
1. Simplify inside the brackets on the right-hand side:
[tex]\[(2 + 2) + 3(2d - 2)\][/tex]
[tex]\[4 + 3(2d - 2)\][/tex]
2. Distribute the 3:
[tex]\[4 + 3 \cdot 2d - 3 \cdot 2\][/tex]
[tex]\[4 + 6d - 6\][/tex]
3. Simplify:
[tex]\[4 + 18d = 4 + 6d - 6\][/tex]
[tex]\[4 + 18d = 6d - 2\][/tex]
4. Subtract 6d from both sides:
[tex]\[4 + 12d = -2\][/tex]
5. Subtract 4 from both sides:
[tex]\[12d = -2 - 4\][/tex]
[tex]\[12d = -6\][/tex]
6. Divide by 12:
[tex]\[d = \frac{-6}{12}\][/tex]
[tex]\[d = -\frac{1}{2}\][/tex]
So the solution is [tex]\(d = -\frac{1}{2}\)[/tex].
### Part (c): [tex]\(4(u + 3) = 5u - 3u\)[/tex]
1. Distribute the 4 on the left-hand side:
[tex]\[4u + 12 = 2u\][/tex]
2. Subtract 2u from both sides:
[tex]\[4u - 2u + 12 = 0\][/tex]
[tex]\[2u + 12 = 0\][/tex]
3. Subtract 12 from both sides:
[tex]\[2u = -12\][/tex]
4. Divide by 2:
[tex]\[u = \frac{-12}{2}\][/tex]
[tex]\[u = -6\][/tex]
So the solution is [tex]\(u = -6\)[/tex].
### Part (d): [tex]\(v + 8 = [(3v + 4 - v) - 2] + 4\)[/tex]
1. Simplify inside the brackets on the right-hand side:
[tex]\[ (3v + 4 - v) - 2 + 4\][/tex]
[tex]\[ (2v + 4) - 2 + 4\][/tex]
[tex]\[ 2v + 8 - 2\][/tex]
[tex]\[ 2v + 6\][/tex]
2. Now, the equation is:
[tex]\[v + 8 = 2v + 6\][/tex]
3. Subtract [tex]\(v\)[/tex] from both sides:
[tex]\[8 = v + 6\][/tex]
4. Subtract 6 from both sides:
[tex]\[v = 2\][/tex]
So the solution is [tex]\(v = 2\)[/tex].
### Part (e): [tex]\(\{[(3 + 4) - 6x \cdot 2] + 10\} = 5\)[/tex]
1. Simplify inside the brackets:
[tex]\[(3 + 4) - 6x \cdot 2\][/tex]
[tex]\[7 - 12x\][/tex]
2. Add the 10:
[tex]\[ (7 - 12x) + 10 = 5\][/tex]
[tex]\[ 17 - 12x = 5\][/tex]
3. Subtract 17 from both sides:
[tex]\[-12x = 5 - 17\][/tex]
[tex]\[-12x = -12\][/tex]
4. Divide by -12:
[tex]\[x = \frac{-12}{-12}\][/tex]
[tex]\[x = 1\][/tex]
So the solution is [tex]\(x = 1\)[/tex].
### Part (f): [tex]\(\{[(7 \cdot 2x) - (7 \cdot 7) + 4x] + 85\} = 0\)[/tex]
1. Simplify inside the brackets:
[tex]\[(7 \cdot 2x) - (7 \cdot 7) + 4x\][/tex]
[tex]\[14x - 49 + 4x\][/tex]
[tex]\[18x - 49\][/tex]
2. Add the 85:
[tex]\[ (18x - 49) + 85 = 0\][/tex]
[tex]\[ 18x + 36 = 0\][/tex]
3. Subtract 36 from both sides:
[tex]\[18x = -36\][/tex]
4. Divide by 18:
[tex]\[x = \frac{-36}{18}\][/tex]
[tex]\[x = -2\][/tex]
So the solution is [tex]\(x = -2\)[/tex].
In summary, the solutions are:
a) [tex]\(x = 1\)[/tex]
b) [tex]\(d = -\frac{1}{2}\)[/tex]
c) [tex]\(u = -6\)[/tex]
d) [tex]\(v = 2\)[/tex]
e) [tex]\(x = 1\)[/tex]
f) [tex]\(x = -2\)[/tex]
