Question

1. $7(2 + 5v) = 3v + 14$
2. $36 - 7p = -7(p - 5)$
3. $-5(-2 - 8m) = 10 + 5m$
4. $-2(v - 2) = -3 - 2v$
5. $30 + 6p = 7(p + 6) - 5$
6. $8x + 38 = -3(-6 - 4x)$
7. $-4(v + 3) = -12v - 4$
8. $-3(v + 4) = 2v - 37$
9. $3 + 5n = 5(n + 2) - 7$
10. $4(k - 8) = -32 + 4k$
11. $-14 - 8x = -2(-3x + 7)$
12. $-(3 - 6b) = 6b + 5$

Explanation:

Let's solve each equation one by one:

Problem 1: \( 7(2 + 5v) = 3v + 14 \)

Step 1: Distribute the 7 on the left side.

\( 14 + 35v = 3v + 14 \)

Step 2: Subtract \( 3v \) from both sides.

\( 14 + 32v = 14 \)

Step 3: Subtract 14 from both sides.

\( 32v = 0 \)

Step 4: Divide both sides by 32.

\( v = 0 \)

Problem 2: \( 36 - 7p = -7(p - 5) \)

Step 1: Distribute the -7 on the right side.

\( 36 - 7p = -7p + 35 \)

Step 2: Add \( 7p \) to both sides.

\( 36 = 35 \)
This is a contradiction, so there is no solution.

Problem 3: \( -5(-2 - 8m) = 10 + 5m \)

Step 1: Distribute the -5 on the left side.

\( 10 + 40m = 10 + 5m \)

Step 2: Subtract \( 5m \) from both sides.

\( 10 + 35m = 10 \)

Step 3: Subtract 10 from both sides.

\( 35m = 0 \)

Step 4: Divide both sides by 35.

\( m = 0 \)

Problem 4: \( -2(v - 2) = -3 - 2v \)

Step 1: Distribute the -2 on the left side.

\( -2v + 4 = -3 - 2v \)

Step 2: Add \( 2v \) to both sides.

\( 4 = -3 \)
This is a contradiction, so there is no solution.

Problem 5: \( 30 + 6p = 7(p + 6) - 5 \)

Step 1: Distribute the 7 on the right side.

\( 30 + 6p = 7p + 42 - 5 \)

Step 2: Simplify the right side.

\( 30 + 6p = 7p + 37 \)

Step 3: Subtract \( 6p \) from both sides.

\( 30 = p + 37 \)

Step 4: Subtract 37 from both sides.

\( p = -7 \)

Problem 6: \( 8x + 38 = -3(-6 - 4x) \)

Step 1: Distribute the -3 on the right side.

\( 8x + 38 = 18 + 12x \)

Step 2: Subtract \( 8x \) from both sides.

\( 38 = 18 + 4x \)

Step 3: Subtract 18 from both sides.

\( 20 = 4x \)

Step 4: Divide both sides by 4.

\( x = 5 \)

Problem 7: \( -4(v + 3) = -12v - 4 \)

Step 1: Distribute the -4 on the left side.

\( -4v - 12 = -12v - 4 \)

Step 2: Add \( 12v \) to both sides.

\( 8v - 12 = -4 \)

Step 3: Add 12 to both sides.

\( 8v = 8 \)

Step 4: Divide both sides by 8.

\( v = 1 \)

Problem 8: \( -3(v + 4) = 2v - 37 \)

Step 1: Distribute the -3 on the left side.

\( -3v - 12 = 2v - 37 \)

Step 2: Subtract \( 2v \) from both sides.

\( -5v - 12 = -37 \)

Step 3: Add 12 to both sides.

\( -5v = -25 \)

Step 4: Divide both sides by -5.

\( v = 5 \)

Problem 9: \( 3 + 5n = 5(n + 2) - 7 \)

Step 1: Distribute the 5 on the right side.

\( 3 + 5n = 5n + 10 - 7 \)

Step 2: Simplify the right side.

\( 3 + 5n = 5n + 3 \)

Step 3: Subtract \( 5n \) from both sides.

\( 3 = 3 \)
This is an identity, so all real numbers are solutions.

Problem 10: \( 4(k - 8) = -32 + 4k \)

Step 1: Distribute the 4 on the left side.

\( 4k - 32 = -32 + 4k \)

Step 2: Subtract \( 4k \) from both sides.

\( -32 = -32 \)
This is an identity, so all real numbers are solutions.

Problem 11: \( -14 - 8x = -2(-3x + 7) \)

Step 1: Distribute the -2 on the right side.

\( -14 - 8x = 6x - 14 \)

Step 2: Add \( 8x \) to both sides.

\( -14 = 14x - 14 \)

Step 3: Add 14 to both sides.

\( 0 = 14x \)

Step 4: Divide both sides by 14.

\( x = 0 \)

Problem 12: \( -(3 - 6b) = 6b + 5 \)

Step 1: Distribute the negative sign on the left side.

\( -3 + 6b = 6b + 5 \)

Step 2: Subtract \( 6b \) from both sides.

\( -3 = 5 \)
This is a contradiction, so there is no solution.

Final Answers:

1. \( \boldsymbol{v = 0} \)
2. No solution
3. \( \boldsymbol{m = 0} \)
4. No solution
5. \( \boldsymbol{p = -7} \)
6. \( \boldsymbol{x = 5} \)
7. \( \boldsymbol{v = 1} \)
8. \( \boldsymbol{v = 5} \)
9. All real numbers
10. All real numbers
11. \( \boldsymbol{x = 0} \)
12. No solution

Answer:

Let's solve each equation one by one:

Problem 1: \( 7(2 + 5v) = 3v + 14 \)

Step 1: Distribute the 7 on the left side.

\( 14 + 35v = 3v + 14 \)

Step 2: Subtract \( 3v \) from both sides.

\( 14 + 32v = 14 \)

Step 3: Subtract 14 from both sides.

\( 32v = 0 \)

Step 4: Divide both sides by 32.

\( v = 0 \)

Problem 2: \( 36 - 7p = -7(p - 5) \)

Step 1: Distribute the -7 on the right side.

\( 36 - 7p = -7p + 35 \)

Step 2: Add \( 7p \) to both sides.

\( 36 = 35 \)
This is a contradiction, so there is no solution.

Problem 3: \( -5(-2 - 8m) = 10 + 5m \)

Step 1: Distribute the -5 on the left side.

\( 10 + 40m = 10 + 5m \)

Step 2: Subtract \( 5m \) from both sides.

\( 10 + 35m = 10 \)

Step 3: Subtract 10 from both sides.

\( 35m = 0 \)

Step 4: Divide both sides by 35.

\( m = 0 \)

Problem 4: \( -2(v - 2) = -3 - 2v \)

Step 1: Distribute the -2 on the left side.

\( -2v + 4 = -3 - 2v \)

Step 2: Add \( 2v \) to both sides.

\( 4 = -3 \)
This is a contradiction, so there is no solution.

Problem 5: \( 30 + 6p = 7(p + 6) - 5 \)

Step 1: Distribute the 7 on the right side.

\( 30 + 6p = 7p + 42 - 5 \)

Step 2: Simplify the right side.

\( 30 + 6p = 7p + 37 \)

Step 3: Subtract \( 6p \) from both sides.

\( 30 = p + 37 \)

Step 4: Subtract 37 from both sides.

\( p = -7 \)

Problem 6: \( 8x + 38 = -3(-6 - 4x) \)

Step 1: Distribute the -3 on the right side.

\( 8x + 38 = 18 + 12x \)

Step 2: Subtract \( 8x \) from both sides.

\( 38 = 18 + 4x \)

Step 3: Subtract 18 from both sides.

\( 20 = 4x \)

Step 4: Divide both sides by 4.

\( x = 5 \)

Problem 7: \( -4(v + 3) = -12v - 4 \)

Step 1: Distribute the -4 on the left side.

\( -4v - 12 = -12v - 4 \)

Step 2: Add \( 12v \) to both sides.

\( 8v - 12 = -4 \)

Step 3: Add 12 to both sides.

\( 8v = 8 \)

Step 4: Divide both sides by 8.

\( v = 1 \)

Problem 8: \( -3(v + 4) = 2v - 37 \)

Step 1: Distribute the -3 on the left side.

\( -3v - 12 = 2v - 37 \)

Step 2: Subtract \( 2v \) from both sides.

\( -5v - 12 = -37 \)

Step 3: Add 12 to both sides.

\( -5v = -25 \)

Step 4: Divide both sides by -5.

\( v = 5 \)

Problem 9: \( 3 + 5n = 5(n + 2) - 7 \)

Step 1: Distribute the 5 on the right side.

\( 3 + 5n = 5n + 10 - 7 \)

Step 2: Simplify the right side.

\( 3 + 5n = 5n + 3 \)

Step 3: Subtract \( 5n \) from both sides.

\( 3 = 3 \)
This is an identity, so all real numbers are solutions.

Problem 10: \( 4(k - 8) = -32 + 4k \)

Step 1: Distribute the 4 on the left side.

\( 4k - 32 = -32 + 4k \)

Step 2: Subtract \( 4k \) from both sides.

\( -32 = -32 \)
This is an identity, so all real numbers are solutions.

Problem 11: \( -14 - 8x = -2(-3x + 7) \)

Step 1: Distribute the -2 on the right side.

\( -14 - 8x = 6x - 14 \)

Step 2: Add \( 8x \) to both sides.

\( -14 = 14x - 14 \)

Step 3: Add 14 to both sides.

\( 0 = 14x \)

Step 4: Divide both sides by 14.

\( x = 0 \)

Problem 12: \( -(3 - 6b) = 6b + 5 \)

Step 1: Distribute the negative sign on the left side.

\( -3 + 6b = 6b + 5 \)

Step 2: Subtract \( 6b \) from both sides.

\( -3 = 5 \)
This is a contradiction, so there is no solution.

Final Answers:

1. \( \boldsymbol{v = 0} \)
2. No solution
3. \( \boldsymbol{m = 0} \)
4. No solution
5. \( \boldsymbol{p = -7} \)
6. \( \boldsymbol{x = 5} \)
7. \( \boldsymbol{v = 1} \)
8. \( \boldsymbol{v = 5} \)
9. All real numbers
10. All real numbers
11. \( \boldsymbol{x = 0} \)
12. No solution