Multi-Step Equation Notes and Worksheets - Lindsay Bowden - Free Printable

Solving Multi-Step Equations Practice

#### Problem 1:
\[ 4(3x + 7) = 64 + 4x \]

Solution:
1. Distribute the 4 on the left side:
\[
4(3x + 7) = 12x + 28
\]
So the equation becomes:
\[
12x + 28 = 64 + 4x
\]

2. Subtract \(4x\) from both sides to get all \(x\)-terms on one side:
\[
12x - 4x + 28 = 64
\]
Simplify:
\[
8x + 28 = 64
\]

3. Subtract 28 from both sides to isolate the \(x\)-term:
\[
8x = 64 - 28
\]
Simplify:
\[
8x = 36
\]

4. Divide both sides by 8 to solve for \(x\):
\[
x = \frac{36}{8} = \frac{9}{2}
\]

Final Answer:
\[
\boxed{\frac{9}{2}}
\]

---

#### Problem 2:
\[ 16 = -4(2x - 5) \]

Solution:
1. Distribute the \(-4\) on the right side:
\[
-4(2x - 5) = -8x + 20
\]
So the equation becomes:
\[
16 = -8x + 20
\]

2. Subtract 20 from both sides to isolate the \(x\)-term:
\[
16 - 20 = -8x
\]
Simplify:
\[
-4 = -8x
\]

3. Divide both sides by \(-8\) to solve for \(x\):
\[
x = \frac{-4}{-8} = \frac{1}{2}
\]

Final Answer:
\[
\boxed{\frac{1}{2}}
\]

---

#### Problem 3:
\[ -8(6 + 5x) = 3x - 5 \]

Solution:
1. Distribute the \(-8\) on the left side:
\[
-8(6 + 5x) = -48 - 40x
\]
So the equation becomes:
\[
-48 - 40x = 3x - 5
\]

2. Add \(40x\) to both sides to get all \(x\)-terms on one side:
\[
-48 = 3x + 40x - 5
\]
Simplify:
\[
-48 = 43x - 5
\]

3. Add 5 to both sides to isolate the \(x\)-term:
\[
-48 + 5 = 43x
\]
Simplify:
\[
-43 = 43x
\]

4. Divide both sides by 43 to solve for \(x\):
\[
x = \frac{-43}{43} = -1
\]

Final Answer:
\[
\boxed{-1}
\]

---

#### Problem 4:
\[ 33 = -3(1 + 4x) \]

Solution:
1. Distribute the \(-3\) on the right side:
\[
-3(1 + 4x) = -3 - 12x
\]
So the equation becomes:
\[
33 = -3 - 12x
\]

2. Add 3 to both sides to isolate the \(x\)-term:
\[
33 + 3 = -12x
\]
Simplify:
\[
36 = -12x
\]

3. Divide both sides by \(-12\) to solve for \(x\):
\[
x = \frac{36}{-12} = -3
\]

Final Answer:
\[
\boxed{-3}
\]

---

#### Problem 5:
\[ \frac{4x - 2}{5} = 0.4 \]

Solution:
1. Eliminate the fraction by multiplying both sides by 5:
\[
5 \cdot \frac{4x - 2}{5} = 5 \cdot 0.4
\]
Simplify:
\[
4x - 2 = 2
\]

2. Add 2 to both sides to isolate the \(x\)-term:
\[
4x = 2 + 2
\]
Simplify:
\[
4x = 4
\]

3. Divide both sides by 4 to solve for \(x\):
\[
x = \frac{4}{4} = 1
\]

Final Answer:
\[
\boxed{1}
\]

---

#### Problem 6:
\[ \frac{2}{3}x - 6 = \frac{1}{3}x + 3 \]

Solution:
1. Eliminate the fractions by multiplying every term by 3 (the least common denominator):
\[
3 \cdot \frac{2}{3}x - 3 \cdot 6 = 3 \cdot \frac{1}{3}x + 3 \cdot 3
\]
Simplify:
\[
2x - 18 = x + 9
\]

2. Subtract \(x\) from both sides to get all \(x\)-terms on one side:
\[
2x - x - 18 = 9
\]
Simplify:
\[
x - 18 = 9
\]

3. Add 18 to both sides to isolate \(x\):
\[
x = 9 + 18
\]
Simplify:
\[
x = 27
\]

Final Answer:
\[
\boxed{27}
\]

---

#### Problem 7:
Find the mistake and correct it.
\[ \frac{1}{4}(12x - 16) = 2x \]

Solution:
1. Distribute the \(\frac{1}{4}\) on the left side:
\[
\frac{1}{4}(12x - 16) = \frac{1}{4} \cdot 12x - \frac{1}{4} \cdot 16 = 3x - 4
\]
So the equation becomes:
\[
3x - 4 = 2x
\]

2. Subtract \(2x\) from both sides to get all \(x\)-terms on one side:
\[
3x - 2x - 4 = 0
\]
Simplify:
\[
x - 4 = 0
\]

3. Add 4 to both sides to solve for \(x\):
\[
x = 4
\]

Mistake in Original Solution:
The original solution likely made an error in distributing or simplifying the terms.

Correct Answer:
\[
\boxed{4}
\]

---

#### Problem 8:
Find the mistake and correct it.
\[ -4 + 14x = -12 + 12x \]

Solution:
1. Subtract \(12x\) from both sides to get all \(x\)-terms on one side:
\[
-4 + 14x - 12x = -12
\]
Simplify:
\[
-4 + 2x = -12
\]

2. Add 4 to both sides to isolate the \(x\)-term:
\[
2x = -12 + 4
\]
Simplify:
\[
2x = -8
\]

3. Divide both sides by 2 to solve for \(x\):
\[
x = \frac{-8}{2} = -4
\]

Mistake in Original Solution:
The original solution likely made an error in combining like terms or isolating \(x\).

Correct Answer:
\[
\boxed{-4}
\]

---

Final Answers:

1. \(\boxed{\frac{9}{2}}\)
2. \(\boxed{\frac{1}{2}}\)
3. \(\boxed{-1}\)
4. \(\boxed{-3}\)
5. \(\boxed{1}\)
6. \(\boxed{27}\)
7. \(\boxed{4}\)
8. \(\boxed{-4}\)