Algebraic Expressions 6 Worksheets - Free Printable

Problem: Solve the given algebraic expressions and identities practice worksheet.

#### Q.1 Add

We will add the given algebraic expressions step by step.

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1. \( t - 8tz, 3tz - z, z - t \)

Add the terms:
\[
(t - 8tz) + (3tz - z) + (z - t)
\]

Combine like terms:
- \( t \)-terms: \( t - t = 0 \)
- \( tz \)-terms: \( -8tz + 3tz = -5tz \)
- \( z \)-terms: \( -z + z = 0 \)

Result:
\[
0 - 5tz + 0 = -5tz
\]

Answer:
\[
\boxed{-5tz}
\]

---

2. \( 7mn + 5, 12mn + 2, 9mn - 8, -2mn - 3 \)

Add the terms:
\[
(7mn + 5) + (12mn + 2) + (9mn - 8) + (-2mn - 3)
\]

Combine like terms:
- \( mn \)-terms: \( 7mn + 12mn + 9mn - 2mn = 26mn \)
- Constant terms: \( 5 + 2 - 8 - 3 = -4 \)

Result:
\[
26mn - 4
\]

Answer:
\[
\boxed{26mn - 4}
\]

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3. \( a + b - 3, b - a + 3, a - b + 3 \)

Add the terms:
\[
(a + b - 3) + (b - a + 3) + (a - b + 3)
\]

Combine like terms:
- \( a \)-terms: \( a - a + a = a \)
- \( b \)-terms: \( b + b - b = b \)
- Constant terms: \( -3 + 3 + 3 = 3 \)

Result:
\[
a + b + 3
\]

Answer:
\[
\boxed{a + b + 3}
\]

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4. \( 14x + 10y - 12xy - 13, 18 - 7x - 10y + 8xy, 4xy \)

Add the terms:
\[
(14x + 10y - 12xy - 13) + (18 - 7x - 10y + 8xy) + 4xy
\]

Combine like terms:
- \( x \)-terms: \( 14x - 7x = 7x \)
- \( y \)-terms: \( 10y - 10y = 0 \)
- \( xy \)-terms: \( -12xy + 8xy + 4xy = 0 \)
- Constant terms: \( -13 + 18 = 5 \)

Result:
\[
7x + 0 + 0 + 5 = 7x + 5
\]

Answer:
\[
\boxed{7x + 5}
\]

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5. \( 5m - 7n, 3n - 4m + 2, 2m - 3mn - 5 \)

Add the terms:
\[
(5m - 7n) + (3n - 4m + 2) + (2m - 3mn - 5)
\]

Combine like terms:
- \( m \)-terms: \( 5m - 4m + 2m = 3m \)
- \( n \)-terms: \( -7n + 3n = -4n \)
- \( mn \)-terms: \( -3mn \)
- Constant terms: \( 2 - 5 = -3 \)

Result:
\[
3m - 4n - 3mn - 3
\]

Answer:
\[
\boxed{3m - 4n - 3mn - 3}
\]

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6. \( 4y(3y^2 + 5y - 7) \) and \( 2(y^3 - 4y^2 + 5) \)

First, expand each expression:
- \( 4y(3y^2 + 5y - 7) = 12y^3 + 20y^2 - 28y \)
- \( 2(y^3 - 4y^2 + 5) = 2y^3 - 8y^2 + 10 \)

Now add the expanded expressions:
\[
(12y^3 + 20y^2 - 28y) + (2y^3 - 8y^2 + 10)
\]

Combine like terms:
- \( y^3 \)-terms: \( 12y^3 + 2y^3 = 14y^3 \)
- \( y^2 \)-terms: \( 20y^2 - 8y^2 = 12y^2 \)
- \( y \)-terms: \( -28y \)
- Constant terms: \( 10 \)

Result:
\[
14y^3 + 12y^2 - 28y + 10
\]

Answer:
\[
\boxed{14y^3 + 12y^2 - 28y + 10}
\]

---

7. \( 5m(3 - m) \) and \( 6m^2 - 13m \)

First, expand \( 5m(3 - m) \):
\[
5m(3 - m) = 15m - 5m^2
\]

Now add the expanded expression to \( 6m^2 - 13m \):
\[
(15m - 5m^2) + (6m^2 - 13m)
\]

Combine like terms:
- \( m^2 \)-terms: \( -5m^2 + 6m^2 = m^2 \)
- \( m \)-terms: \( 15m - 13m = 2m \)

Result:
\[
m^2 + 2m
\]

Answer:
\[
\boxed{m^2 + 2m}
\]

---

8. \( 7xy + 5yz - 3zx, 4yz + 9zx - 4y, -3zx + 5y - 2xy \)

Add the terms:
\[
(7xy + 5yz - 3zx) + (4yz + 9zx - 4y) + (-3zx + 5y - 2xy)
\]

Combine like terms:
- \( xy \)-terms: \( 7xy - 2xy = 5xy \)
- \( yz \)-terms: \( 5yz + 4yz = 9yz \)
- \( zx \)-terms: \( -3zx + 9zx - 3zx = 3zx \)
- \( y \)-terms: \( -4y + 5y = y \)

Result:
\[
5xy + 9yz + 3zx + y
\]

Answer:
\[
\boxed{5xy + 9yz + 3zx + y}
\]

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Q.2 Simplify

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1. \( (a + b)(2a - 3b + c) - (2a - 3b)c \)

First, expand \( (a + b)(2a - 3b + c) \):
\[
(a + b)(2a - 3b + c) = a(2a - 3b + c) + b(2a - 3b + c)
\]
\[
= 2a^2 - 3ab + ac + 2ab - 3b^2 + bc
\]
\[
= 2a^2 - ab + ac - 3b^2 + bc
\]

Next, expand \( (2a - 3b)c \):
\[
(2a - 3b)c = 2ac - 3bc
\]

Now subtract:
\[
(2a^2 - ab + ac - 3b^2 + bc) - (2ac - 3bc)
\]
\[
= 2a^2 - ab + ac - 3b^2 + bc - 2ac + 3bc
\]

Combine like terms:
- \( a^2 \)-terms: \( 2a^2 \)
- \( ab \)-terms: \( -ab \)
- \( ac \)-terms: \( ac - 2ac = -ac \)
- \( b^2 \)-terms: \( -3b^2 \)
- \( bc \)-terms: \( bc + 3bc = 4bc \)

Result:
\[
2a^2 - ab - ac - 3b^2 + 4bc
\]

Answer:
\[
\boxed{2a^2 - ab - ac - 3b^2 + 4bc}
\]

---

2. \( (x + y)(2x + y) + (x + 2y)(x - y) \)

First, expand \( (x + y)(2x + y) \):
\[
(x + y)(2x + y) = x(2x + y) + y(2x + y)
\]
\[
= 2x^2 + xy + 2xy + y^2
\]
\[
= 2x^2 + 3xy + y^2
\]

Next, expand \( (x + 2y)(x - y) \):
\[
(x + 2y)(x - y) = x(x - y) + 2y(x - y)
\]
\[
= x^2 - xy + 2xy - 2y^2
\]
\[
= x^2 + xy - 2y^2
\]

Now add the two results:
\[
(2x^2 + 3xy + y^2) + (x^2 + xy - 2y^2)
\]

Combine like terms:
- \( x^2 \)-terms: \( 2x^2 + x^2 = 3x^2 \)
- \( xy \)-terms: \( 3xy + xy = 4xy \)
- \( y^2 \)-terms: \( y^2 - 2y^2 = -y^2 \)

Result:
\[
3x^2 + 4xy - y^2
\]

Answer:
\[
\boxed{3x^2 + 4xy - y^2}
\]

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3. \( (a + b + c)(a + b - c) \)

Use the identity \( (p + q)(p - q) = p^2 - q^2 \), where \( p = a + b \) and \( q = c \):
\[
(a + b + c)(a + b - c) = (a + b)^2 - c^2
\]

Expand \( (a + b)^2 \):
\[
(a + b)^2 = a^2 + 2ab + b^2
\]

So:
\[
(a + b + c)(a + b - c) = a^2 + 2ab + b^2 - c^2
\]

Answer:
\[
\boxed{a^2 + 2ab + b^2 - c^2}
\]

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Q.3 Subtract \( 5x^2 - 4y^2 + 6y - 3 \) from \( 7x^2 - 4xy + 8y^2 + 5x - 3y \)

Subtract:
\[
(7x^2 - 4xy + 8y^2 + 5x - 3y) - (5x^2 - 4y^2 + 6y - 3)
\]

Distribute the negative sign:
\[
= 7x^2 - 4xy + 8y^2 + 5x - 3y - 5x^2 + 4y^2 - 6y + 3
\]

Combine like terms:
- \( x^2 \)-terms: \( 7x^2 - 5x^2 = 2x^2 \)
- \( xy \)-terms: \( -4xy \)
- \( y^2 \)-terms: \( 8y^2 + 4y^2 = 12y^2 \)
- \( x \)-terms: \( 5x \)
- \( y \)-terms: \( -3y - 6y = -9y \)
- Constant terms: \( 3 \)

Result:
\[
2x^2 - 4xy + 12y^2 + 5x - 9y + 3
\]

Answer:
\[
\boxed{2x^2 - 4xy + 12y^2 + 5x - 9y + 3}
\]

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Q.4 Subtract \( 4a - 7ab + 3b + 12 \) from \( 12a - 9ab + 5b - 3 \)

Subtract:
\[
(12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12)
\]

Distribute the negative sign:
\[
= 12a - 9ab + 5b - 3 - 4a + 7ab - 3b - 12
\]

Combine like terms:
- \( a \)-terms: \( 12a - 4a = 8a \)
- \( ab \)-terms: \( -9ab + 7ab = -2ab \)
- \( b \)-terms: \( 5b - 3b = 2b \)
- Constant terms: \( -3 - 12 = -15 \)

Result:
\[
8a - 2ab + 2b - 15
\]

Answer:
\[
\boxed{8a - 2ab + 2b - 15}
\]

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Q.5 Subtract \( 3xy + 5yz - 7zx \) from \( 5xy - 2yz - 2zx + 10xyz \)

Subtract:
\[
(5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx)
\]

Distribute the negative sign:
\[
= 5xy - 2yz - 2zx + 10xyz - 3xy - 5yz + 7zx
\]

Combine like terms:
- \( xy \)-terms: \( 5xy - 3xy = 2xy \)
- \( yz \)-terms: \( -2yz - 5yz = -7yz \)
- \( zx \)-terms: \( -2zx + 7zx = 5zx \)
- \( xyz \)-terms: \( 10xyz \)

Result:
\[
2xy - 7yz + 5zx + 10xyz
\]

Answer:
\[
\boxed{2xy - 7yz + 5zx + 10xyz}
\]

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Q.6 Subtract \( 24ab - 10b - 18a \) from \( 30ab + 12b + 14a \)

Subtract:
\[
(30ab + 12b + 14a) - (24ab - 10b - 18a)
\]

Distribute the negative sign:
\[
= 30ab + 12b + 14a - 24ab + 10b + 18a
\]

Combine like terms:
- \( ab \)-terms: \( 30ab - 24ab = 6ab \)
- \( b \)-terms: \( 12b + 10b = 22b \)
- \( a \)-terms: \( 14a + 18a = 32a \)

Result:
\[
6ab + 22b + 32a
\]

Answer:
\[
\boxed{6ab + 22b + 32a}
\]

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Final Answers:

1. \(\boxed{-5tz}\)
2. \(\boxed{26mn - 4}\)
3. \(\boxed{a + b + 3}\)
4. \(\boxed{7x + 5}\)
5. \(\boxed{3m - 4n - 3mn - 3}\)
6. \(\boxed{14y^3 + 12y^2 - 28y + 10}\)
7. \(\boxed{m^2 + 2m}\)
8. \(\boxed{5xy + 9yz + 3zx + y}\)
9. \(\boxed{2a^2 - ab - ac - 3b^2 + 4bc}\)
10. \(\boxed{3x^2 + 4xy - y^2}\)
11. \(\boxed{a^2 + 2ab + b^2 - c^2}\)
12. \(\boxed{2x^2 - 4xy + 12y^2 + 5x - 9y + 3}\)
13. \(\boxed{8a - 2ab + 2b - 15}\)
14. \(\boxed{2xy - 7yz + 5zx + 10xyz}\)
15. \(\boxed{6ab + 22b + 32a}\)