Solving multi step equations | PDF - Free Printable

Problem Description:

The task involves solving a series of multi-step equations. Each equation requires simplification, isolating the variable, and solving for the variable. Some equations may have no solution or infinitely many solutions.

Solution Approach:

1. Simplify both sides of the equation by combining like terms.
2. Isolate the variable by performing inverse operations (e.g., addition/subtraction, multiplication/division).
3. Check for special cases:
- If the variable cancels out and you are left with a true statement (e.g., \( 5 = 5 \)), the equation has infinitely many solutions.
- If the variable cancels out and you are left with a false statement (e.g., \( 5 = 8 \)), the equation has no solution.

Step-by-Step Solutions:

#### 1. \( -20 = -4x + 6x \)
- Combine like terms on the right side: \( -4x + 6x = 2x \).
- Equation becomes: \( -20 = 2x \).
- Divide both sides by 2: \( x = -10 \).

Answer: \( x = -10 \)

#### 2. \( 6 = 1 - 2x + 5 \)
- Combine constants on the right side: \( 1 + 5 = 6 \).
- Equation becomes: \( 6 = 6 - 2x \).
- Subtract 6 from both sides: \( 0 = -2x \).
- Divide by -2: \( x = 0 \).

Answer: \( x = 0 \)

#### 3. \( 8x - 2 = -9 + 7x \)
- Subtract \( 7x \) from both sides: \( 8x - 7x - 2 = -9 \).
- Simplify: \( x - 2 = -9 \).
- Add 2 to both sides: \( x = -7 \).

Answer: \( x = -7 \)

#### 4. \( a + 5 = -3a + 5 \)
- Subtract 5 from both sides: \( a = -3a \).
- Add \( 3a \) to both sides: \( 4a = 0 \).
- Divide by 4: \( a = 0 \).

Answer: \( a = 0 \)

#### 5. \( 4n + 8 = 6n \)
- Subtract \( 4n \) from both sides: \( 8 = 2n \).
- Divide by 2: \( n = 4 \).

Answer: \( n = 4 \)

#### 6. \( p + 7 = 2p + 8 \)
- Subtract \( p \) from both sides: \( 7 = p + 8 \).
- Subtract 8 from both sides: \( -1 = p \).

Answer: \( p = -1 \)

#### 7. \( 5p - 14 = 8p + 4 \)
- Subtract \( 5p \) from both sides: \( -14 = 3p + 4 \).
- Subtract 4 from both sides: \( -18 = 3p \).
- Divide by 3: \( p = -6 \).

Answer: \( p = -6 \)

#### 8. \( p - 8 = -9 + p \)
- Subtract \( p \) from both sides: \( -8 = -9 \).
- This is a false statement, so there is no solution.

Answer: No solution

#### 9. \( 8 + x = -6(x + 4) \)
- Distribute on the right side: \( 8 + x = -6x - 24 \).
- Add \( 6x \) to both sides: \( 8 + 7x = -24 \).
- Subtract 8 from both sides: \( 7x = -32 \).
- Divide by 7: \( x = -\frac{32}{7} \).

Answer: \( x = -\frac{32}{7} \)

#### 10. \( 12 = -4(d - 3) \)
- Distribute on the right side: \( 12 = -4d + 12 \).
- Subtract 12 from both sides: \( 0 = -4d \).
- Divide by -4: \( d = 0 \).

Answer: \( d = 0 \)

#### 11. \( 14 = -v - v + 8 \)
- Combine like terms on the right side: \( -v - v = -2v \).
- Equation becomes: \( 14 = -2v + 8 \).
- Subtract 8 from both sides: \( 6 = -2v \).
- Divide by -2: \( v = -3 \).

Answer: \( v = -3 \)

#### 12. \( -(7 - 4x) = 9 \)
- Distribute the negative sign: \( -7 + 4x = 9 \).
- Add 7 to both sides: \( 4x = 16 \).
- Divide by 4: \( x = 4 \).

Answer: \( x = 4 \)

#### 13. \( 18 = -4(6 + u) + 3(u - 2) \)
- Distribute on both terms: \( 18 = -24 - 4u + 3u - 6 \).
- Combine like terms: \( 18 = -30 - u \).
- Add 30 to both sides: \( 48 = -u \).
- Multiply by -1: \( u = -48 \).

Answer: \( u = -48 \)

#### 14. \( 4x + 34 = -2(7 - x) \)
- Distribute on the right side: \( 4x + 34 = -14 + 2x \).
- Subtract \( 2x \) from both sides: \( 2x + 34 = -14 \).
- Subtract 34 from both sides: \( 2x = -48 \).
- Divide by 2: \( x = -24 \).

Answer: \( x = -24 \)

#### 15. \( 2(k - 3) = k + 8 + 2k \)
- Distribute on the left side: \( 2k - 6 = k + 8 + 2k \).
- Combine like terms on the right side: \( 2k - 6 = 3k + 8 \).
- Subtract \( 2k \) from both sides: \( -6 = k + 8 \).
- Subtract 8 from both sides: \( -14 = k \).

Answer: \( k = -14 \)

#### 16. \( -(1 + 7x) + 6(-7 - x) = 36 \)
- Distribute on both terms: \( -1 - 7x - 42 - 6x = 36 \).
- Combine like terms: \( -43 - 13x = 36 \).
- Add 43 to both sides: \( -13x = 79 \).
- Divide by -13: \( x = -\frac{79}{13} \).

Answer: \( x = -\frac{79}{13} \)

#### 17. \( -(3x - 4) + 4 = 6(3x + 1) + 43 \)
- Distribute on both sides: \( -3x + 4 + 4 = 18x + 6 + 43 \).
- Simplify: \( -3x + 8 = 18x + 49 \).
- Subtract 8 from both sides: \( -3x = 18x + 41 \).
- Subtract \( 18x \) from both sides: \( -21x = 41 \).
- Divide by -21: \( x = -\frac{41}{21} \).

Answer: \( x = -\frac{41}{21} \)

#### 18. \( -5(1 - 2x) + 3(8 - 2x) = -4x + 8x \)
- Distribute on both sides: \( -5 + 10x + 24 - 6x = 4x \).
- Combine like terms: \( 19 + 4x = 4x \).
- Subtract \( 4x \) from both sides: \( 19 = 0 \).
- This is a false statement, so there is no solution.

Answer: No solution

#### 19. \( 3m - 22 = -8(m - 6) \)
- Distribute on the right side: \( 3m - 22 = -8m + 48 \).
- Add \( 8m \) to both sides: \( 11m - 22 = 48 \).
- Add 22 to both sides: \( 11m = 70 \).
- Divide by 11: \( m = \frac{70}{11} \).

Answer: \( m = \frac{70}{11} \)

#### 20. \( -5(1 - 2x) + 3(8 - 2x) = -4x + 8x \)
- Already solved in step 18. The equation has no solution.

Answer: No solution

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & x = -10 \\
2. & x = 0 \\
3. & x = -7 \\
4. & a = 0 \\
5. & n = 4 \\
6. & p = -1 \\
7. & p = -6 \\
8. & \text{No solution} \\
9. & x = -\frac{32}{7} \\
10. & d = 0 \\
11. & v = -3 \\
12. & x = 4 \\
13. & u = -48 \\
14. & x = -24 \\
15. & k = -14 \\
16. & x = -\frac{79}{13} \\
17. & x = -\frac{41}{21} \\
18. & \text{No solution} \\
19. & m = \frac{70}{11} \\
20. & \text{No solution}
\end{array}
}
\]