Algebra practice problems involving equations and word problems.

Let's solve each problem step-by-step. I'll go through all the equations and word problems one by one, showing the solution process clearly.

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6. $\frac{P}{4} + 7 = 5$

Subtract 7 from both sides:
$$
\frac{P}{4} = 5 - 7 = -2
$$
Multiply both sides by 4:
$$
P = -8
$$

✔ Answer: $P = -8$

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7. $6z + 10 = -2$

Subtract 10:
$$
6z = -12
$$
Divide by 6:
$$
z = -2
$$

✔ Answer: $z = -2$

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8. $7 - 5x = 5 - 7x$

Move variables to one side, constants to the other:
Add $7x$ to both sides:
$$
7 + 2x = 5
$$
Subtract 7:
$$
2x = -2 \Rightarrow x = -1
$$

✔ Answer: $x = -1$

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9. $2y + 3 = 5y + 7$

Subtract $2y$ and 7:
$$
-4 = 3y \Rightarrow y = -\frac{4}{3}
$$

✔ Answer: $y = -\frac{4}{3}$

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10. $5x - \frac{7}{3} = 3x - \frac{2}{3}$

Subtract $3x$:
$$
2x - \frac{7}{3} = -\frac{2}{3}
$$
Add $\frac{7}{3}$:
$$
2x = \frac{5}{3} \Rightarrow x = \frac{5}{6}
$$

✔ Answer: $x = \frac{5}{6}$

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11. $\frac{3}{2}z - \frac{5}{3} = \frac{5}{3} + \frac{7}{2}z$

Bring like terms together:
Subtract $\frac{7}{2}z$ and add $\frac{5}{3}$ to both sides:
$$
\frac{3}{2}z - \frac{7}{2}z = \frac{5}{3} + \frac{5}{3}
\Rightarrow -2z = \frac{10}{3}
\Rightarrow z = -\frac{5}{3}
$$

✔ Answer: $z = -\frac{5}{3}$

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12. $4y = \frac{2}{3}(3y - 7)$

Distribute:
$$
4y = 2y - \frac{14}{3}
$$
Subtract $2y$:
$$
2y = -\frac{14}{3} \Rightarrow y = -\frac{7}{3}
$$

✔ Answer: $y = -\frac{7}{3}$

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13. $\frac{2x - 3}{3} = 1 - \frac{2}{3}x$

Multiply both sides by 3:
$$
2x - 3 = 3 - 2x
$$
Add $2x$ and 3:
$$
4x = 6 \Rightarrow x = \frac{3}{2}
$$

✔ Answer: $x = \frac{3}{2}$

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14. $5x + 3 = \frac{4}{3}(1 + x)$

Multiply both sides by 3:
$$
15x + 9 = 4(1 + x) = 4 + 4x
$$
Subtract $4x$ and 4:
$$
11x + 5 = 0 \Rightarrow x = -\frac{5}{11}
$$

✔ Answer: $x = -\frac{5}{11}$

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15. $\frac{5}{3}(2y - 1) = (3y - 5)$

Multiply both sides by 3:
$$
5(2y - 1) = 3(3y - 5)
\Rightarrow 10y - 5 = 9y - 15
\Rightarrow y = -10
$$

✔ Answer: $y = -10$

---

16. $\frac{2}{5}(2x - 1) = \frac{4}{5}(3x - 5)$

Multiply both sides by 5:
$$
2(2x - 1) = 4(3x - 5)
\Rightarrow 4x - 2 = 12x - 20
\Rightarrow -8x = -18 \Rightarrow x = \frac{9}{4}
$$

✔ Answer: $x = \frac{9}{4}$

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17. $\frac{2y + 5}{3} = 3y - 8$

Multiply both sides by 3:
$$
2y + 5 = 9y - 24
\Rightarrow -7y = -29 \Rightarrow y = \frac{29}{7}
$$

✔ Answer: $y = \frac{29}{7}$

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18. $\frac{3x}{8} = 27$

Multiply both sides by 8:
$$
3x = 216 \Rightarrow x = 72
$$

✔ Answer: $x = 72$

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19. $\frac{y}{9} - \frac{y}{12} = \frac{1}{108}$

Find common denominator: LCM of 9 and 12 is 36.
$$
\frac{4y - 3y}{36} = \frac{1}{108} \Rightarrow \frac{y}{36} = \frac{1}{108}
$$
Cross-multiply:
$$
108y = 36 \Rightarrow y = \frac{1}{3}
$$

✔ Answer: $y = \frac{1}{3}$

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20. $y - 2 = \frac{1}{5}(3y - 1)$

Multiply both sides by 5:
$$
5y - 10 = 3y - 1
\Rightarrow 2y = 9 \Rightarrow y = \frac{9}{2}
$$

✔ Answer: $y = \frac{9}{2}$

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21. $\frac{7P + 8}{9} = 9$

Multiply both sides by 9:
$$
7P + 8 = 81 \Rightarrow 7P = 73 \Rightarrow P = \frac{73}{7}
$$

✔ Answer: $P = \frac{73}{7}$

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22. $5x + 0.7 = 2.9$

Subtract 0.7:
$$
5x = 2.2 \Rightarrow x = 0.44
$$

✔ Answer: $x = 0.44$

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23. $4x + 0.8 = 7.2$

Subtract 0.8:
$$
4x = 6.4 \Rightarrow x = 1.6
$$

✔ Answer: $x = 1.6$

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24. $0.16(5x - 2) = 0.4(x + 1)$

Expand both sides:
$$
0.8x - 0.32 = 0.4x + 0.4
$$
Subtract $0.4x$ and add 0.32:
$$
0.4x = 0.72 \Rightarrow x = 1.8
$$

✔ Answer: $x = 1.8$

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25. $2.8y = 54 + y$

Subtract $y$:
$$
1.8y = 54 \Rightarrow y = \frac{54}{1.8} = 30
$$

✔ Answer: $y = 30$

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26. If $x = y + 2$, find $y$ from: $y - \frac{x - 2}{2} = \frac{2}{3}$

Substitute $x = y + 2$:
$$
y - \frac{(y + 2) - 2}{2} = \frac{2}{3}
\Rightarrow y - \frac{y}{2} = \frac{2}{3}
\Rightarrow \frac{y}{2} = \frac{2}{3} \Rightarrow y = \frac{4}{3}
$$

✔ Answer: $y = \frac{4}{3}$

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27. If $x = y - 3$, find $y$ from: $y - \frac{x - y}{3} = \frac{4}{5}(y - x)$

Substitute $x = y - 3$:

Left side:
$$
y - \frac{(y - 3) - y}{3} = y - \frac{-3}{3} = y + 1
$$

Right side:
$$
\frac{4}{5}(y - (y - 3)) = \frac{4}{5}(3) = \frac{12}{5}
$$

So:
$$
y + 1 = \frac{12}{5} \Rightarrow y = \frac{12}{5} - 1 = \frac{7}{5}
$$

✔ Answer: $y = \frac{7}{5}$

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28. If $x = y - 2$ and $\frac{x}{y} - \frac{x + 1}{y} = 1$, find $x$ and $y$

Simplify the equation:
$$
\frac{x - (x + 1)}{y} = 1 \Rightarrow \frac{-1}{y} = 1 \Rightarrow y = -1
$$
Then $x = y - 2 = -3$

Check:
$$
\frac{-3}{-1} - \frac{-3 + 1}{-1} = 3 - \frac{-2}{-1} = 3 - 2 = 1 ✔
$$

✔ Answer: $x = -3$, $y = -1$

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29. Find 3 consecutive numbers whose sum is 45.

Let them be $x, x+1, x+2$

Sum:
$$
x + x+1 + x+2 = 3x + 3 = 45 \Rightarrow 3x = 42 \Rightarrow x = 14
$$

Numbers: 14, 15, 16

✔ Answer: 14, 15, 16

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30. Find 3 odd consecutive numbers whose sum is 27.

Let them be $x, x+2, x+4$ (odd numbers)

Sum:
$$
3x + 6 = 27 \Rightarrow 3x = 21 \Rightarrow x = 7
$$

Numbers: 7, 9, 11

✔ Answer: 7, 9, 11

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31. If the smaller of two consecutive odd integers is doubled, the result is 7 more than the larger integer.

Let smaller = $x$, larger = $x+2$

Given:
$$
2x = (x + 2) + 7 = x + 9
\Rightarrow 2x - x = 9 \Rightarrow x = 9
$$

Integers: 9 and 11

Check: $2 \times 9 = 18$, $11 + 7 = 18$ ✔

✔ Answer: 9 and 11

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32. Nine added to thrice a whole number gives 45.

Let number = $x$

$$
3x + 9 = 45 \Rightarrow 3x = 36 \Rightarrow x = 12
$$

✔ Answer: 12

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33. Twice a number when decreased by 7 gives 15.

$$
2x - 7 = 15 \Rightarrow 2x = 22 \Rightarrow x = 11
$$

✔ Answer: 11

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34. A number is $\frac{2}{5}$ times another number. If their sum is 70, find the numbers.

Let first number = $x$, second = $\frac{2}{5}x$

Sum:
$$
x + \frac{2}{5}x = \frac{7}{5}x = 70 \Rightarrow x = 70 \times \frac{5}{7} = 50
$$

Other number: $\frac{2}{5} \times 50 = 20$

✔ Answer: 50 and 20

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35. A number when added to its half gives 72.

Let number = $x$

$$
x + \frac{1}{2}x = \frac{3}{2}x = 72 \Rightarrow x = 72 \times \frac{2}{3} = 48
$$

✔ Answer: 48

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36. A number is as much greater than 21 as it is less than 71.

Let number = $x$

So: $x - 21 = 71 - x$

Solve:
$$
2x = 92 \Rightarrow x = 46
$$

Check: $46 - 21 = 25$, $71 - 46 = 25$ ✔

✔ Answer: 46

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 6 | $P = -8$ |
| 7 | $z = -2$ |
| 8 | $x = -1$ |
| 9 | $y = -\frac{4}{3}$ |
| 10 | $x = \frac{5}{6}$ |
| 11 | $z = -\frac{5}{3}$ |
| 12 | $y = -\frac{7}{3}$ |
| 13 | $x = \frac{3}{2}$ |
| 14 | $x = -\frac{5}{11}$ |
| 15 | $y = -10$ |
| 16 | $x = \frac{9}{4}$ |
| 17 | $y = \frac{29}{7}$ |
| 18 | $x = 72$ |
| 19 | $y = \frac{1}{3}$ |
| 20 | $y = \frac{9}{2}$ |
| 21 | $P = \frac{73}{7}$ |
| 22 | $x = 0.44$ |
| 23 | $x = 1.6$ |
| 24 | $x = 1.8$ |
| 25 | $y = 30$ |
| 26 | $y = \frac{4}{3}$ |
| 27 | $y = \frac{7}{5}$ |
| 28 | $x = -3$, $y = -1$ |
| 29 | 14, 15, 16 |
| 30 | 7, 9, 11 |
| 31 | 9, 11 |
| 32 | 12 |
| 33 | 11 |
| 34 | 50 and 20 |
| 35 | 48 |
| 36 | 46 |

Let me know if you'd like these in a formatted document or need explanations for any specific ones!