Collection of algebra exercises focusing on substitution, simplification, and geometric calculations for students.

Let's solve each problem one by one with clear explanations.

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11. Subtract $24ab - 10b + 18a$ from $26ab - 8b - 7a$.

We are subtracting the second expression from the first:

$$
(26ab - 8b - 7a) - (24ab - 10b + 18a)
$$

Distribute the negative sign:

$$
= 26ab - 8b - 7a - 24ab + 10b - 18a
$$

Now combine like terms:

- $26ab - 24ab = 2ab$
- $-8b + 10b = 2b$
- $-7a - 18a = -25a$

Answer: $2ab + 2b - 25a$

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12. What should be added to $x^2 + y^2$ to get $2x^2 + 3xy$?

Let the unknown expression be $A$. Then:

$$
x^2 + y^2 + A = 2x^2 + 3xy
$$

Solve for $A$:

$$
A = (2x^2 + 3xy) - (x^2 + y^2) = 2x^2 + 3xy - x^2 - y^2 = x^2 + 3xy - y^2
$$

Answer: $x^2 + 3xy - y^2$

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13. Find the value of $10x - 2$ when $x = -2$.

Substitute $x = -2$:

$$
10(-2) - 2 = -20 - 2 = -22
$$

Answer: $-22$

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14. Find the value of $5x^2 + 4x - 2$ when $x = -2$.

$$
5(-2)^2 + 4(-2) - 2 = 5(4) - 8 - 2 = 20 - 8 - 2 = 10
$$

Answer: $10$

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15. Find the value of $x^2 - 3x + 5$ when $x = 2$.

$$
(2)^2 - 3(2) + 5 = 4 - 6 + 5 = 3
$$

Answer: $3$

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16. Find the value of $7a - 4b$ when $a = 2$, $b = -2$.

$$
7(2) - 4(-2) = 14 + 8 = 22
$$

Answer: $22$

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17. Evaluate $\frac{x}{5} - \frac{y}{5}$ when $x = 2$, $y = -3$.

$$
\frac{2}{5} - \frac{-3}{5} = \frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1
$$

Answer: $1$

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18. If $Z = 5$, find the value of $2 - 3(4 - Z)$.

$$
2 - 3(4 - 5) = 2 - 3(-1) = 2 + 3 = 5
$$

Answer: $5$

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19. If $P = -6$, find the value of $\frac{2P^2}{3} - 1$.

First, $P^2 = (-6)^2 = 36$

$$
\frac{2 \cdot 36}{3} - 1 = \frac{72}{3} - 1 = 24 - 1 = 23
$$

Answer: $23$

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20. If $a = 2$, find the value of $a\left(1 + \frac{1}{a}\right)$.

$$
2\left(1 + \frac{1}{2}\right) = 2\left(\frac{3}{2}\right) = 3
$$

Answer: $3$

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21. Evaluate: $2a^2 + 3b^2 + 1c^2$ if $a = 1$, $b = 0$, $c = -4$.

$$
2(1)^2 + 3(0)^2 + 1(-4)^2 = 2(1) + 0 + 1(16) = 2 + 0 + 16 = 18
$$

Answer: $18$

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22. Find the value of $(a - b)(a + b) - (2a - b)(a + b)$ if $a = 3$, $b = 2$.

First, compute each part:

- $a - b = 3 - 2 = 1$
- $a + b = 3 + 2 = 5$
- $2a - b = 6 - 2 = 4$

Now:

$$
(1)(5) - (4)(5) = 5 - 20 = -15
$$

Answer: $-15$

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23. If $a + b = 5$, find the value of $\frac{1}{2}a + \frac{1}{2}b$.

Factor:

$$
\frac{1}{2}(a + b) = \frac{1}{2}(5) = 2.5
$$

Answer: $2.5$ or $\frac{5}{2}$

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24. Find the value of $3(2a + b)$ if $a = 2$, $b = -1$.

$$
2a + b = 4 - 1 = 3 \\
3(3) = 9
$$

Answer: $9$

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25. Find the value of $5x + 3$ if $x = 9$.

$$
5(9) + 3 = 45 + 3 = 48
$$

Answer: $48$

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26. What should be added to $x^2 + 2x$ to obtain $x^2 + 9$?

Let the expression be $A$:

$$
x^2 + 2x + A = x^2 + 9 \\
A = (x^2 + 9) - (x^2 + 2x) = 9 - 2x
$$

Answer: $9 - 2x$

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27. What should be added to $x^2 + x + a$ to obtain $x^2 + x - 3$?

Let $A$ be the expression:

$$
x^2 + x + a + A = x^2 + x - 3 \\
A = (x^2 + x - 3) - (x^2 + x + a) = -3 - a
$$

Answer: $-3 - a$

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28. What should be subtracted from $3a + 7b - 16$ to get $2a + 8b$?

Let $A$ be the expression to subtract:

$$
(3a + 7b - 16) - A = 2a + 8b \\
A = (3a + 7b - 16) - (2a + 8b) = a - b - 16
$$

Answer: $a - b - 16$

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29. If $a = 3$, $b = -3$, find the value of $a^2 + b^2$.

$$
3^2 + (-3)^2 = 9 + 9 = 18
$$

Answer: $18$

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30. If $a = 5$, $b = -5$, find the value of $a^2 - b^2$.

$$
5^2 - (-5)^2 = 25 - 25 = 0
$$

Answer: $0$

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31. Simplify the following expression if $x = 8$: $5x + 6(x - 3)$

First simplify algebraically:

$$
5x + 6x - 18 = 11x - 18
$$

Now substitute $x = 8$:

$$
11(8) - 18 = 88 - 18 = 70
$$

Answer: $70$

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32. If a side of an equilateral triangle is $7x$ units, find the perimeter.

All sides equal → Perimeter = $3 \times 7x = 21x$

Answer: $21x$ units

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33. Find the perimeter of a square whose side is $4a$ cm.

Perimeter of square = $4 \times \text{side} = 4 \times 4a = 16a$

Answer: $16a$ cm

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34. Find the perimeter of a regular pentagon whose side is $4x$ cm.

Regular pentagon has 5 equal sides → $5 \times 4x = 20x$

Answer: $20x$ cm

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35. Side of a square is $2x$ cm. Find its area.

Area = side² = $(2x)^2 = 4x^2$

Answer: $4x^2$ cm²

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36. Base of a triangle is $2x$ units and height is $3y$ units. Find the area.

Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height}$

$$
= \frac{1}{2} \times 2x \times 3y = x \times 3y = 3xy
$$

Answer: $3xy$ square units

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✔ Final Answers Summary:

| Q | Answer |
|---|--------|
| 11 | $2ab + 2b - 25a$ |
| 12 | $x^2 + 3xy - y^2$ |
| 13 | $-22$ |
| 14 | $10$ |
| 15 | $3$ |
| 16 | $22$ |
| 17 | $1$ |
| 18 | $5$ |
| 19 | $23$ |
| 20 | $3$ |
| 21 | $18$ |
| 22 | $-15$ |
| 23 | $2.5$ or $\frac{5}{2}$ |
| 24 | $9$ |
| 25 | $48$ |
| 26 | $9 - 2x$ |
| 27 | $-3 - a$ |
| 28 | $a - b - 16$ |
| 29 | $18$ |
| 30 | $0$ |
| 31 | $70$ |
| 32 | $21x$ |
| 33 | $16a$ |
| 34 | $20x$ |
| 35 | $4x^2$ |
| 36 | $3xy$ |

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