Math worksheet with multi-step equations and answer key for variables on both sides.

I will solve this multi-step equations worksheet by going through each problem one by one, showing the steps to isolate the variable and find its value. I'll then match the answer to the correct color as instructed.

Let me start with problem 1.

Problem 1: 4x - 7 = 2x + 5

Step 1: Subtract 2x from both sides to get all x terms on one side.
4x - 2x - 7 = 5
2x - 7 = 5

Step 2: Add 7 to both sides to isolate the term with x.
2x = 5 + 7
2x = 12

Step 3: Divide both sides by 2.
x = 12 / 2
x = 6

The answer is x = 6, which corresponds to the color Red.

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Problem 2: 20k + 5 = 5k + 65

Step 1: Subtract 5k from both sides.
20k - 5k + 5 = 65
15k + 5 = 65

Step 2: Subtract 5 from both sides.
15k = 65 - 5
15k = 60

Step 3: Divide both sides by 15.
k = 60 / 15
k = 4

The answer is k = 4, which corresponds to the color Orange.

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Problem 3: -9y = 20 + y

Step 1: Subtract y from both sides to get all y terms on one side.
-9y - y = 20
-10y = 20

Step 2: Divide both sides by -10.
y = 20 / (-10)
y = -2

The answer is y = -2, which corresponds to the color Purple.

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Problem 4: 5x - 1/4 = 3x - 5/4

Step 1: Subtract 3x from both sides.
5x - 3x - 1/4 = -5/4
2x - 1/4 = -5/4

Step 2: Add 1/4 to both sides.
2x = -5/4 + 1/4
2x = -4/4
2x = -1

Step 3: Divide both sides by 2.
x = -1 / 2
x = -1/2

The answer is x = -1/2, which corresponds to the color Blue.

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Problem 5: 16k - 5 = 13k

Step 1: Subtract 13k from both sides.
16k - 13k - 5 = 0
3k - 5 = 0

Step 2: Add 5 to both sides.
3k = 5

Step 3: Divide both sides by 3.
k = 5/3

The answer is k = 5/3, which corresponds to the color Orange.

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Problem 6: -x - 7 = 13 - x

Wait, let's look at this carefully. If we add x to both sides:
-7 = 13

This is a contradiction! This equation has no solution.

However, looking at the provided answer choices in the image, there seems to be an error in the problem or the answer key. The problem as written has no solution, but the answer key shows "x = -10" and "x = 20". Let me re-examine the problem.

Looking again at the image, the problem is actually:
6. -x - 7 = 13 - x

But if we add x to both sides, we get -7 = 13, which is impossible. This suggests there might be a typo in the worksheet. However, since the worksheet provides answers, I will assume the intended problem was different or proceed with the given answer.

Given the answer key shows "x = -10" for Green and "x = 20" for Orange, and since the problem as written has no solution, I will note this discrepancy.

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Problem 7: 18 + 45m = 6m + 12

Step 1: Subtract 6m from both sides.
18 + 45m - 6m = 12
18 + 39m = 12

Step 2: Subtract 18 from both sides.
39m = 12 - 18
39m = -6

Step 3: Divide both sides by 39.
m = -6 / 39
m = -2/13 (simplified)

The answer is m = -2/13, which corresponds to the color Green.

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Problem 8: 3(x + 7) = 6(x + 2)

Step 1: Distribute on both sides.
3x + 21 = 6x + 12

Step 2: Subtract 3x from both sides.
21 = 3x + 12

Step 3: Subtract 12 from both sides.
9 = 3x

Step 4: Divide both sides by 3.
x = 3

The answer is x = 3, which corresponds to the color Red.

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Problem 9: 8x - 6 = 5x + 12

Step 1: Subtract 5x from both sides.
8x - 5x - 6 = 12
3x - 6 = 12

Step 2: Add 6 to both sides.
3x = 18

Step 3: Divide both sides by 3.
x = 6

The answer is x = 6, which corresponds to the color Yellow.

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Problem 10: -2(w + 10) = 6w + 12

Step 1: Distribute on the left side.
-2w - 20 = 6w + 12

Step 2: Add 2w to both sides.
-20 = 8w + 12

Step 3: Subtract 12 from both sides.
-32 = 8w

Step 4: Divide both sides by 8.
w = -32 / 8
w = -4

The answer is w = -4, which corresponds to the color Blue.

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Summary of Answers:

1. x = 6 → Red
2. k = 4 → Orange
3. y = -2 → Purple
4. x = -1/2 → Blue
5. k = 5/3 → Orange
6. No solution (problem may have typo) → Based on answer key, perhaps x = -10 → Green or x = 20 → Orange
7. m = -2/13 → Green
8. x = 3 → Red
9. x = 6 → Yellow
10. w = -4 → Blue

Note: Problem 6 appears to have an error as written, since it leads to a contradiction. The intended problem might have been different.