Simplifying and Solving Equations worksheet with 20 algebra problems for practice.

Since the user has not specified any of the options (ExplainAnswer, TeachTopic, GeneratePractice) as true, and only asked to “solve the problem accurately,” I will interpret this as solving all 20 equations listed in the worksheet.

I’ll go equation by equation, solve for the variable, and verify each step carefully.

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Equation 1:
2(3 + A) = -5A
→ Expand left: 6 + 2A = -5A
→ Add 5A to both sides: 6 + 7A = 0
→ Subtract 6: 7A = -6
→ Divide by 7: A = -6/7

✔ Check: Left = 2(3 - 6/7) = 2*(15/7) = 30/7; Right = -5*(-6/7) = 30/7 → Correct.

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Equation 2:
-2(3x - 4) = 8
→ Expand: -6x + 8 = 8
→ Subtract 8: -6x = 0
→ Divide by -6: x = 0

✔ Check: Left = -2(0 - 4) = -2*(-4) = 8 → Correct.

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Equation 3:
7 + 9z = 7z + 8
→ Subtract 7z: 7 + 2z = 8
→ Subtract 7: 2z = 1
→ Divide by 2: z = 1/2

✔ Check: Left = 7 + 9*(1/2) = 7 + 4.5 = 11.5; Right = 7*(1/2) + 8 = 3.5 + 8 = 11.5 → Correct.

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Equation 4:
3(-y + 4) = -5(y + 1)
→ Expand: -3y + 12 = -5y - 5
→ Add 5y: 2y + 12 = -5
→ Subtract 12: 2y = -17
→ Divide by 2: y = -17/2

✔ Check: Left = 3(-(-17/2)+4) = 3(17/2 + 8/2) = 3*(25/2) = 75/2
Right = -5(-17/2 + 1) = -5(-17/2 + 2/2) = -5*(-15/2) = 75/2 → Correct.

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Equation 5:
-2(4 + z) = -2(4 + y)
→ Expand both: -8 - 2z = -8 - 2y
→ Add 8: -2z = -2y
→ Divide by -2: z = y

This is an identity — infinitely many solutions where z equals y. But since it’s asking to determine value, perhaps they want relationship? Or maybe typo? Assuming we solve for one variable in terms of other: z = y

But if forced to give numerical answer, not possible. However, looking at context, likely expects z = y or infinite solutions. But let's see others first.

Actually, re-read: “determine the value of the unknown” — but here two variables. Maybe misprint? Let’s assume they meant same variable? Unlikely. Perhaps leave as z = y.

But wait — maybe it’s a trick? If you divide both sides by -2: 4+z = 4+y → z=y. So no unique solution. We’ll note that.

For now, write: Infinitely many solutions; z = y

But since worksheet probably expects single answers, maybe skip? No — let’s keep going.

Actually, looking back, maybe it’s intentional. We’ll record: z = y

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Equation 6:
3(1 + x) = -9x + 6
→ Expand: 3 + 3x = -9x + 6
→ Add 9x: 3 + 12x = 6
→ Subtract 3: 12x = 3
→ Divide by 12: x = 3/12 = 1/4

✔ Check: Left = 3(1 + 0.25) = 3*1.25 = 3.75; Right = -9*(0.25) + 6 = -2.25 + 6 = 3.75 → Correct.

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Equation 7:
-7 + 4a = 5(a + 4)
→ Expand right: -7 + 4a = 5a + 20
→ Subtract 4a: -7 = a + 20
→ Subtract 20: -27 = a → a = -27

✔ Check: Left = -7 + 4*(-27) = -7 -108 = -115; Right = 5(-27+4)=5*(-23)=-115 → Correct.

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Equation 8:
1 + 10b = 8b + 9
→ Subtract 8b: 1 + 2b = 9
→ Subtract 1: 2b = 8
→ Divide by 2: b = 4

✔ Check: Left = 1 + 40 = 41; Right = 32 + 9 = 41 → Correct.

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Equation 9:
3 + 1 = 2(2n - 1)
→ Simplify left: 4 = 2(2n - 1)
→ Divide both sides by 2: 2 = 2n - 1
→ Add 1: 3 = 2n
→ Divide by 2: n = 3/2

✔ Check: Right = 2(2*(3/2) -1) = 2(3-1)=2*2=4 → Matches left → Correct.

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Equation 10:
2x + 6c = 5x + 1
Wait — two variables again? x and c. Probably typo? Or maybe treat as expression? But instruction says “determine value of unknown”. Likely error.

Looking at pattern, maybe should be same variable? Or perhaps c is constant? Not clear.

Assume it’s meant to be solved for x in terms of c? Then:

2x + 6c = 5x + 1
→ Subtract 2x: 6c = 3x + 1
→ Subtract 1: 6c - 1 = 3x
→ Divide by 3: x = (6c - 1)/3 = 2c - 1/3

But messy. Alternatively, maybe it’s 2x + 6 = 5x + 1? That would make sense.

Check original image description? Since I can’t see image, rely on text given.

In user input, it says: "10. 2x + 6c = 5x + 1"

Perhaps c is a constant? Still ambiguous.

To proceed, I’ll assume it’s a typo and should be numbers. But better to solve as is.

So final answer: x = 2c - 1/3

But let’s flag this.

Alternatively, maybe “c” is meant to be “x”? Try:

If 2x + 6x = 5x + 1 → 8x = 5x + 1 → 3x=1 → x=1/3 — nice number.

Or 2x + 6 = 5x + 1 → 5 = 3x → x=5/3.

Given ambiguity, I’ll stick with literal: x = 2c - 1/3

But for consistency, perhaps the worksheet intends single-variable equations. Looking ahead, Equation 11 has d, etc.

Actually, Equation 11: 5s - 2 = s + 8 — single variable.

Equation 12: 6g - 4 = 6g - 9 — also single.

So Equation 10 might have typo. But since I must solve as given, I’ll do so.

Final: x = 2c - 1/3

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Equation 11:
5s - 2 = s + 8
→ Subtract s: 4s - 2 = 8
→ Add 2: 4s = 10
→ Divide by 4: s = 10/4 = 5/2

✔ Check: Left = 5*(2.5) - 2 = 12.5 - 2 = 10.5; Right = 2.5 + 8 = 10.5 → Correct.

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Equation 12:
6g - 4 = 6g - 9
→ Subtract 6g: -4 = -9 → Contradiction!

No solution. No solution

✔ Check: Impossible unless -4 = -9, which is false.

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Equation 13:
-2 - 4m = 7m + 8
→ Add 4m: -2 = 11m + 8
→ Subtract 8: -10 = 11m
→ Divide by 11: m = -10/11

✔ Check: Left = -2 -4*(-10/11) = -2 + 40/11 = (-22 + 40)/11 = 18/11
Right = 7*(-10/11) + 8 = -70/11 + 88/11 = 18/11 → Correct.

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Equation 14:
-6h - 4k = -9
Two variables again. Similar issue.

Solve for h in terms of k:
-6h = 4k - 9
→ h = (9 - 4k)/6

Or k in terms of h:
-4k = 6h + 9 → k = -(6h + 9)/4

Not solvable uniquely. Assume typo? Or accept parametric.

But look at next equations — most are single variable.

Perhaps “k” is meant to be “h”? Try:

-6h -4h = -9 → -10h = -9 → h=9/10 — nice.

Or maybe constant? I’ll solve literally.

h = (9 - 4k)/6

But to match format, perhaps write as fraction.

Final: h = \frac{9 - 4k}{6}

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Equation 15:
-6(1 - n) = -2n
→ Expand: -6 + 6n = -2n
→ Add 2n: -6 + 8n = 0
→ Add 6: 8n = 6
→ Divide by 8: n = 6/8 = 3/4

✔ Check: Left = -6(1 - 0.75) = -6(0.25) = -1.5; Right = -2*(0.75) = -1.5 → Correct.

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Equation 16:
-2(r - 3) = -2(r + 1)
→ Expand: -2r + 6 = -2r - 2
→ Add 2r: 6 = -2 → Contradiction!

No solution

✔ Check: 6 ≠ -2 always.

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Equation 17:
2 + t = 7 - 6t
→ Add 6t: 2 + 7t = 7
→ Subtract 2: 7t = 5
→ Divide by 7: t = 5/7

✔ Check: Left = 2 + 5/7 = 19/7; Right = 7 - 30/7 = 49/7 - 30/7 = 19/7 → Correct.

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Equation 18:
-6f + 7 = -2f + 5
→ Add 6f: 7 = 4f + 5
→ Subtract 5: 2 = 4f
→ Divide by 4: f = 2/4 = 1/2

✔ Check: Left = -6*(0.5) + 7 = -3 + 7 = 4; Right = -2*(0.5) + 5 = -1 + 5 = 4 → Correct.

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Equation 19:
-6k + 1 = -2 + 7k
→ Add 6k: 1 = -2 + 13k
→ Add 2: 3 = 13k
→ Divide by 13: k = 3/13

✔ Check: Left = -6*(3/13) + 1 = -18/13 + 13/13 = -5/13
Right = -2 + 21/13 = -26/13 + 21/13 = -5/13 → Correct.

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Equation 20:
-6k + 1 = -2 + 7k — Wait, this is same as 19? In user input, it says:

“19. -6k + 1 = -2 + 7k”
“20. -6k + 1 = -2 + 7k” — duplicate?

Probably typo in transcription. Looking back at user message:

"19. -6k + 1 = -2 + 7k
20. -6k + 1 = -2 + 7k"

Yes, identical. So either mistake or intentional.

We already solved 19: k=3/13

So 20 is same: k = 3/13

But perhaps in original image, different? Since I must use given text, I'll solve as is.

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Now, compiling all answers:

1. A = -6/7
2. x = 0
3. z = 1/2
4. y = -17/2
5. z = y (infinitely many solutions)
6. x = 1/4
7. a = -27
8. b = 4
9. n = 3/2
10. x = 2c - 1/3 (or if typo, possibly different)
11. s = 5/2
12. No solution
13. m = -10/11
14. h = (9 - 4k)/6
15. n = 3/4
16. No solution
17. t = 5/7
18. f = 1/2
19. k = 3/13
20. k = 3/13 (same as 19)

Note: Equations 5, 10, 14 involve multiple variables or identities. For those, we provide relationships or state no unique solution.

But since the worksheet likely expects numerical answers, and some may have typos, I’ll present the solved values as per calculation.

For final output, list all 20 answers clearly.

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Final Answer:
1. \( A = -\frac{6}{7} \)
2. \( x = 0 \)
3. \( z = \frac{1}{2} \)
4. \( y = -\frac{17}{2} \)
5. \( z = y \) (infinitely many solutions)
6. \( x = \frac{1}{4} \)
7. \( a = -27 \)
8. \( b = 4 \)
9. \( n = \frac{3}{2} \)
10. \( x = 2c - \frac{1}{3} \)
11. \( s = \frac{5}{2} \)
12. No solution
13. \( m = -\frac{10}{11} \)
14. \( h = \frac{9 - 4k}{6} \)
15. \( n = \frac{3}{4} \)
16. No solution
17. \( t = \frac{5}{7} \)
18. \( f = \frac{1}{2} \)
19. \( k = \frac{3}{13} \)
20. \( k = \frac{3}{13} \)