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

Here is the complete solution for all 8 problems on the "Solving Multi-Step Equations Practice" worksheet, with detailed step-by-step work shown for each.

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Problem 1:
Equation: `4(3x + 7) = 64 + 4x`

Solution:

1. Distribute the 4 on the left side:
`4 * 3x + 4 * 7 = 64 + 4x`
`12x + 28 = 64 + 4x`

2. Subtract 4x from both sides to get variable terms on one side:
`12x - 4x + 28 = 64 + 4x - 4x`
`8x + 28 = 64`

3. Subtract 28 from both sides to isolate the variable term:
`8x + 28 - 28 = 64 - 28`
`8x = 36`

4. Divide both sides by 8 to solve for x:
`8x / 8 = 36 / 8`
`x = 36/8`

5. Simplify the fraction:
`x = 9/2` or `x = 4.5`

Answer: `x = 9/2` (or `4.5`)

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Problem 2:
Equation: `16 = -4(2x - 5)`

Solution:

1. Distribute the -4 on the right side:
`16 = -4 * 2x + (-4) * (-5)`
`16 = -8x + 20`

2. Subtract 20 from both sides:
`16 - 20 = -8x + 20 - 20`
`-4 = -8x`

3. Divide both sides by -8:
`-4 / -8 = -8x / -8`
`1/2 = x`

Answer: `x = 1/2`

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Problem 3:
Equation: `-8(6 + 5x) = 3x - 5`

Solution:

1. Distribute the -8 on the left side:
`-8 * 6 + (-8) * 5x = 3x - 5`
`-48 - 40x = 3x - 5`

2. Add 40x to both sides to get variable terms on one side:
`-48 - 40x + 40x = 3x - 5 + 40x`
`-48 = 43x - 5`

3. Add 5 to both sides to isolate the variable term:
`-48 + 5 = 43x - 5 + 5`
`-43 = 43x`

4. Divide both sides by 43:
`-43 / 43 = 43x / 43`
`-1 = x`

Answer: `x = -1`

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Problem 4:
Equation: `33 = -3(1 + 4x)`

Solution:

1. Distribute the -3 on the right side:
`33 = -3 * 1 + (-3) * 4x`
`33 = -3 - 12x`

2. Add 3 to both sides:
`33 + 3 = -3 - 12x + 3`
`36 = -12x`

3. Divide both sides by -12:
`36 / -12 = -12x / -12`
`-3 = x`

Answer: `x = -3`

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Problem 5:
Equation: `(4x - 2)/5 = 0.4`

Solution:

1. Multiply both sides by 5 to eliminate the denominator:
`5 * (4x - 2)/5 = 5 * 0.4`
`4x - 2 = 2`

2. Add 2 to both sides:
`4x - 2 + 2 = 2 + 2`
`4x = 4`

3. Divide both sides by 4:
`4x / 4 = 4 / 4`
`x = 1`

Answer: `x = 1`

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Problem 6:
Equation: `(2/3)x - 6 = (1/3)x + 3`

Solution:

1. Subtract (1/3)x from both sides to get variable terms on one side:
`(2/3)x - (1/3)x - 6 = (1/3)x - (1/3)x + 3`
`(1/3)x - 6 = 3`

2. Add 6 to both sides:
`(1/3)x - 6 + 6 = 3 + 6`
`(1/3)x = 9`

3. Multiply both sides by 3 to solve for x:
`3 * (1/3)x = 3 * 9`
`x = 27`

Answer: `x = 27`

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Problem 7: Find the mistake and correct it.
Equation: `(1/4)(12x - 16) = 2x`

Mistake Analysis:
The most common mistake here is forgetting to distribute the `1/4` to *both* terms inside the parentheses. A student might incorrectly simplify this to `3x - 16 = 2x`, which is wrong because they only multiplied `1/4` by `12x` and ignored multiplying it by `-16`.

Correct Solution:

1. Distribute the `1/4` to both terms inside the parentheses:
`(1/4)*12x - (1/4)*16 = 2x`
`3x - 4 = 2x`

2. Subtract 2x from both sides:
`3x - 2x - 4 = 2x - 2x`
`x - 4 = 0`

3. Add 4 to both sides:
`x - 4 + 4 = 0 + 4`
`x = 4`

Correction: The mistake was not distributing `1/4` to the `-16`. The correct solution is `x = 4`.

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Problem 8: Find the mistake and correct it.
Equation: `-4 + 14x = -12 + 12x`

Mistake Analysis:
This equation doesn't have a distribution error. The mistake is likely in the solving process. A common error is misaligning terms when moving variables or constants. For example, a student might subtract `12x` from the left and add `12x` to the right, or mishandle the negative signs.

Correct Solution:

1. Subtract 12x from both sides to get variable terms on one side:
`-4 + 14x - 12x = -12 + 12x - 12x`
`-4 + 2x = -12`

2. Add 4 to both sides to isolate the variable term:
`-4 + 4 + 2x = -12 + 4`
`2x = -8`

3. Divide both sides by 2:
`2x / 2 = -8 / 2`
`x = -4`

Correction: The mistake was likely an error in combining like terms or handling negative numbers during the solving steps. The correct solution is `x = -4`.

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Final Answers Summary:

1. `x = 9/2` or `4.5`
2. `x = 1/2`
3. `x = -1`
4. `x = -3`
5. `x = 1`
6. `x = 27`
7. Mistake: Did not distribute `1/4` to `-16`. Correct answer: `x = 4`
8. Mistake: Likely an error in combining terms/negatives. Correct answer: `x = -4`