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Step-by-step solution for: Algebraic equations - Free worksheets, PowerPoints and other ...
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Show Answer Key & Explanations
Step-by-step solution for: Algebraic equations - Free worksheets, PowerPoints and other ...
Let’s solve each equation one by one. We’ll isolate the variable (like x, y, t, etc.) on one side of the equals sign by doing the opposite operation to both sides.
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1. x + 5 = 9
Subtract 5 from both sides:
x = 9 - 5 → x = 4
2. x + 7 = 15
Subtract 7:
x = 15 - 7 → x = 8
3. x + 1 = 8
Subtract 1:
x = 8 - 1 → x = 7
4. x + 11 = 21
Subtract 11:
x = 21 - 11 → x = 10
5. x + 6 = 4
Subtract 6:
x = 4 - 6 → x = -2
6. x + 4 = 4
Subtract 4:
x = 4 - 4 → x = 0
7. y + 20 = 32
Subtract 20:
y = 32 - 20 → y = 12
8. y + 12 = 30
Subtract 12:
y = 30 - 12 → y = 18
9. t + 8 = 101
Subtract 8:
t = 101 - 8 → t = 93
10. r + 0.5 = 9
Subtract 0.5:
r = 9 - 0.5 → r = 8.5
11. x - 1 = 10
Add 1:
x = 10 + 1 → x = 11
12. x - 5 = 2
Add 5:
x = 2 + 5 → x = 7
13. x - 8 = 20
Add 8:
x = 20 + 8 → x = 28
14. x - 9 = -3
Add 9:
x = -3 + 9 → x = 6
15. x - 3 = 21
Add 3:
x = 21 + 3 → x = 24
16. x - 7 = 4
Add 7:
x = 4 + 7 → x = 11
17. y - 10 = 34
Add 10:
y = 34 + 10 → y = 44
18. t - 52 = 11
Add 52:
t = 11 + 52 → t = 63
19. t - 3.5 = 9
Add 3.5:
t = 9 + 3.5 → t = 12.5
20. p - 21 = -4
Add 21:
p = -4 + 21 → p = 17
21. 2x = 12
Divide by 2:
x = 12 ÷ 2 → x = 6
22. 5x = 15
Divide by 5:
x = 15 ÷ 5 → x = 3
23. 3x = -6
Divide by 3:
x = -6 ÷ 3 → x = -2
24. 8x = 32
Divide by 8:
x = 32 ÷ 8 → x = 4
25. 4x = 14
Divide by 4:
x = 14 ÷ 4 → x = 3.5
26. 9x = -27
Divide by 9:
x = -27 ÷ 9 → x = -3
27. 6x = 31
Divide by 6:
x = 31 ÷ 6 → x ≈ 5.166... or as fraction x = 31/6
But since this is likely expecting decimal or exact value — we’ll write it as x = 31/6 unless told otherwise. But let’s check if they want decimals? Looking at others, some are decimals. Let’s keep as fraction for accuracy, but maybe simplify? 31 and 6 have no common factors → x = 31/6
Wait — actually, in school problems like this, sometimes they expect decimal. 31 ÷ 6 = 5.166..., which is repeating. But perhaps leave as fraction? Let me see problem 29: 3y=17 → same thing. I think we should give exact values. So:
→ x = 31/6
But let’s double-check instructions — “solve accurately”. Fraction is more accurate. Okay.
Actually, looking ahead — problem 30 has 3.5p = 10.5 → that’s decimal. Maybe they accept fractions or decimals. To be safe, I’ll use decimals where easy, fractions when needed.
But 31/6 is fine. Alternatively, mixed number? No, better improper fraction or decimal. Since 31÷6=5.166..., and it repeats, better to write as fraction.
I’ll go with x = 31/6
But wait — let’s look at problem 25: 4x=14 → x=3.5, which is decimal. So maybe they prefer decimals. Hmm.
Actually, 31/6 is approximately 5.1667, but since it’s not terminating, perhaps leave as fraction. I think for consistency, I’ll use fractions when not whole numbers, unless decimal is clean.
But let’s proceed and decide per problem.
For now: x = 31/6
28. 11x = 88
Divide by 11:
x = 88 ÷ 11 → x = 8
29. 3y = 17
Divide by 3:
y = 17 ÷ 3 → y = 17/3 (or ≈5.666...)
Again, fraction is exact → y = 17/3
30. 3.5p = 10.5
Divide both sides by 3.5:
p = 10.5 ÷ 3.5
Calculate: 10.5 ÷ 3.5 = ?
Well, 3.5 × 3 = 10.5 → so p = 3
31. 2x + 1 = 13
First subtract 1:
2x = 12
Then divide by 2:
x = 6 → x = 6
32. 5x - 2 = 23
Add 2:
5x = 25
Divide by 5:
x = 5 → x = 5
33. 8t + 3 = 43
Subtract 3:
8t = 40
Divide by 8:
t = 5 → t = 5
34. 4x + 10 = 42
Subtract 10:
4x = 32
Divide by 4:
x = 8 → x = 8
35. 7x - 2 = 65
Add 2:
7x = 67
Divide by 7:
x = 67/7 → x = 67/7 (≈9.571...)
Exact value: x = 67/7
36. 3y + 4 = -11
Subtract 4:
3y = -15
Divide by 3:
y = -5 → y = -5
37. 6t + 7 = 28
Subtract 7:
6t = 21
Divide by 6:
t = 21/6 = 7/2 → t = 3.5 or 7/2
Since 21÷6=3.5, and it’s exact decimal → t = 3.5
38. 7x - 1 = 25
Add 1:
7x = 26
Divide by 7:
x = 26/7 → x = 26/7 (≈3.714...)
Exact: x = 26/7
39. 3p - 4 = -2
Add 4:
3p = 2
Divide by 3:
p = 2/3 → p = 2/3
40. 12q + 3 = 36
Subtract 3:
12q = 33
Divide by 12:
q = 33/12 = 11/4 → q = 11/4 or 2.75
Since 33÷12=2.75 exactly → q = 2.75
Now, let’s list all answers clearly.
Note: For fractions that can be simplified, we did. For decimals that terminate, we used them. For non-terminating, we kept fractions.
Final Answers:
1. x = 4
2. x = 8
3. x = 7
4. x = 10
5. x = -2
6. x = 0
7. y = 12
8. y = 18
9. t = 93
10. r = 8.5
11. x = 11
12. x = 7
13. x = 28
14. x = 6
15. x = 24
16. x = 11
17. y = 44
18. t = 63
19. t = 12.5
20. p = 17
21. x = 6
22. x = 3
23. x = -2
24. x = 4
25. x = 3.5
26. x = -3
27. x = 31/6
28. x = 8
29. y = 17/3
30. p = 3
31. x = 6
32. x = 5
33. t = 5
34. x = 8
35. x = 67/7
36. y = -5
37. t = 3.5
38. x = 26/7
39. p = 2/3
40. q = 2.75
Double-checked calculations — all seem correct.
Final Answer:
1. x = 4
2. x = 8
3. x = 7
4. x = 10
5. x = -2
6. x = 0
7. y = 12
8. y = 18
9. t = 93
10. r = 8.5
11. x = 11
12. x = 7
13. x = 28
14. x = 6
15. x = 24
16. x = 11
17. y = 44
18. t = 63
19. t = 12.5
20. p = 17
21. x = 6
22. x = 3
23. x = -2
24. x = 4
25. x = 3.5
26. x = -3
27. x = 31/6
28. x = 8
29. y = 17/3
30. p = 3
31. x = 6
32. x = 5
33. t = 5
34. x = 8
35. x = 67/7
36. y = -5
37. t = 3.5
38. x = 26/7
39. p = 2/3
40. q = 2.75
---
1. x + 5 = 9
Subtract 5 from both sides:
x = 9 - 5 → x = 4
2. x + 7 = 15
Subtract 7:
x = 15 - 7 → x = 8
3. x + 1 = 8
Subtract 1:
x = 8 - 1 → x = 7
4. x + 11 = 21
Subtract 11:
x = 21 - 11 → x = 10
5. x + 6 = 4
Subtract 6:
x = 4 - 6 → x = -2
6. x + 4 = 4
Subtract 4:
x = 4 - 4 → x = 0
7. y + 20 = 32
Subtract 20:
y = 32 - 20 → y = 12
8. y + 12 = 30
Subtract 12:
y = 30 - 12 → y = 18
9. t + 8 = 101
Subtract 8:
t = 101 - 8 → t = 93
10. r + 0.5 = 9
Subtract 0.5:
r = 9 - 0.5 → r = 8.5
11. x - 1 = 10
Add 1:
x = 10 + 1 → x = 11
12. x - 5 = 2
Add 5:
x = 2 + 5 → x = 7
13. x - 8 = 20
Add 8:
x = 20 + 8 → x = 28
14. x - 9 = -3
Add 9:
x = -3 + 9 → x = 6
15. x - 3 = 21
Add 3:
x = 21 + 3 → x = 24
16. x - 7 = 4
Add 7:
x = 4 + 7 → x = 11
17. y - 10 = 34
Add 10:
y = 34 + 10 → y = 44
18. t - 52 = 11
Add 52:
t = 11 + 52 → t = 63
19. t - 3.5 = 9
Add 3.5:
t = 9 + 3.5 → t = 12.5
20. p - 21 = -4
Add 21:
p = -4 + 21 → p = 17
21. 2x = 12
Divide by 2:
x = 12 ÷ 2 → x = 6
22. 5x = 15
Divide by 5:
x = 15 ÷ 5 → x = 3
23. 3x = -6
Divide by 3:
x = -6 ÷ 3 → x = -2
24. 8x = 32
Divide by 8:
x = 32 ÷ 8 → x = 4
25. 4x = 14
Divide by 4:
x = 14 ÷ 4 → x = 3.5
26. 9x = -27
Divide by 9:
x = -27 ÷ 9 → x = -3
27. 6x = 31
Divide by 6:
x = 31 ÷ 6 → x ≈ 5.166... or as fraction x = 31/6
But since this is likely expecting decimal or exact value — we’ll write it as x = 31/6 unless told otherwise. But let’s check if they want decimals? Looking at others, some are decimals. Let’s keep as fraction for accuracy, but maybe simplify? 31 and 6 have no common factors → x = 31/6
Wait — actually, in school problems like this, sometimes they expect decimal. 31 ÷ 6 = 5.166..., which is repeating. But perhaps leave as fraction? Let me see problem 29: 3y=17 → same thing. I think we should give exact values. So:
→ x = 31/6
But let’s double-check instructions — “solve accurately”. Fraction is more accurate. Okay.
Actually, looking ahead — problem 30 has 3.5p = 10.5 → that’s decimal. Maybe they accept fractions or decimals. To be safe, I’ll use decimals where easy, fractions when needed.
But 31/6 is fine. Alternatively, mixed number? No, better improper fraction or decimal. Since 31÷6=5.166..., and it repeats, better to write as fraction.
I’ll go with x = 31/6
But wait — let’s look at problem 25: 4x=14 → x=3.5, which is decimal. So maybe they prefer decimals. Hmm.
Actually, 31/6 is approximately 5.1667, but since it’s not terminating, perhaps leave as fraction. I think for consistency, I’ll use fractions when not whole numbers, unless decimal is clean.
But let’s proceed and decide per problem.
For now: x = 31/6
28. 11x = 88
Divide by 11:
x = 88 ÷ 11 → x = 8
29. 3y = 17
Divide by 3:
y = 17 ÷ 3 → y = 17/3 (or ≈5.666...)
Again, fraction is exact → y = 17/3
30. 3.5p = 10.5
Divide both sides by 3.5:
p = 10.5 ÷ 3.5
Calculate: 10.5 ÷ 3.5 = ?
Well, 3.5 × 3 = 10.5 → so p = 3
31. 2x + 1 = 13
First subtract 1:
2x = 12
Then divide by 2:
x = 6 → x = 6
32. 5x - 2 = 23
Add 2:
5x = 25
Divide by 5:
x = 5 → x = 5
33. 8t + 3 = 43
Subtract 3:
8t = 40
Divide by 8:
t = 5 → t = 5
34. 4x + 10 = 42
Subtract 10:
4x = 32
Divide by 4:
x = 8 → x = 8
35. 7x - 2 = 65
Add 2:
7x = 67
Divide by 7:
x = 67/7 → x = 67/7 (≈9.571...)
Exact value: x = 67/7
36. 3y + 4 = -11
Subtract 4:
3y = -15
Divide by 3:
y = -5 → y = -5
37. 6t + 7 = 28
Subtract 7:
6t = 21
Divide by 6:
t = 21/6 = 7/2 → t = 3.5 or 7/2
Since 21÷6=3.5, and it’s exact decimal → t = 3.5
38. 7x - 1 = 25
Add 1:
7x = 26
Divide by 7:
x = 26/7 → x = 26/7 (≈3.714...)
Exact: x = 26/7
39. 3p - 4 = -2
Add 4:
3p = 2
Divide by 3:
p = 2/3 → p = 2/3
40. 12q + 3 = 36
Subtract 3:
12q = 33
Divide by 12:
q = 33/12 = 11/4 → q = 11/4 or 2.75
Since 33÷12=2.75 exactly → q = 2.75
Now, let’s list all answers clearly.
Note: For fractions that can be simplified, we did. For decimals that terminate, we used them. For non-terminating, we kept fractions.
Final Answers:
1. x = 4
2. x = 8
3. x = 7
4. x = 10
5. x = -2
6. x = 0
7. y = 12
8. y = 18
9. t = 93
10. r = 8.5
11. x = 11
12. x = 7
13. x = 28
14. x = 6
15. x = 24
16. x = 11
17. y = 44
18. t = 63
19. t = 12.5
20. p = 17
21. x = 6
22. x = 3
23. x = -2
24. x = 4
25. x = 3.5
26. x = -3
27. x = 31/6
28. x = 8
29. y = 17/3
30. p = 3
31. x = 6
32. x = 5
33. t = 5
34. x = 8
35. x = 67/7
36. y = -5
37. t = 3.5
38. x = 26/7
39. p = 2/3
40. q = 2.75
Double-checked calculations — all seem correct.
Final Answer:
1. x = 4
2. x = 8
3. x = 7
4. x = 10
5. x = -2
6. x = 0
7. y = 12
8. y = 18
9. t = 93
10. r = 8.5
11. x = 11
12. x = 7
13. x = 28
14. x = 6
15. x = 24
16. x = 11
17. y = 44
18. t = 63
19. t = 12.5
20. p = 17
21. x = 6
22. x = 3
23. x = -2
24. x = 4
25. x = 3.5
26. x = -3
27. x = 31/6
28. x = 8
29. y = 17/3
30. p = 3
31. x = 6
32. x = 5
33. t = 5
34. x = 8
35. x = 67/7
36. y = -5
37. t = 3.5
38. x = 26/7
39. p = 2/3
40. q = 2.75
Parent Tip: Review the logic above to help your child master the concept of basic algebraic expression worksheet.
