Practice worksheet for Class 8 students focusing on algebraic expressions and identities, including addition, simplification, and subtraction problems.
Class 8 Algebraic Expressions and Identities Practice Worksheet with questions on addition, simplification, and subtraction of algebraic terms.
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Show Answer Key & Explanations
Step-by-step solution for: Simplifying Expressions Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Simplifying Expressions Worksheets - Math Monks
Here are the step-by-step solutions for the problems in your worksheet.
To add algebraic expressions, we group "like terms" together (terms with the same variables and powers) and then add their numbers (coefficients).
1. $t - 8tz$, $3tz - z$, $z - t$
* Group like terms: $(t - t) + (-8tz + 3tz) + (-z + z)$
* Simplify: $0 - 5tz + 0$
* Answer: $-5tz$
2. $7mn + 5$, $12mn + 2$, $9mn - 8$, $-2mn - 3$
* Group $mn$ terms: $7mn + 12mn + 9mn - 2mn = 26mn$
* Group numbers: $5 + 2 - 8 - 3 = 7 - 11 = -4$
* Answer: $26mn - 4$
3. $a + b - 3$, $b - a + 3$, $a - b + 3$
* Group $a$ terms: $a - a + a = a$
* Group $b$ terms: $b + b - b = b$
* Group numbers: $-3 + 3 + 3 = 3$
* Answer: $a + b + 3$
4. $14x + 10y - 12xy - 13$, $18 - 7x - 10y + 8xy$, $4xy$
* Group $x$ terms: $14x - 7x = 7x$
* Group $y$ terms: $10y - 10y = 0$
* Group $xy$ terms: $-12xy + 8xy + 4xy = 0$
* Group numbers: $-13 + 18 = 5$
* Answer: $7x + 5$
5. $5m - 7n$, $3n - 4m + 2$, $2m - 3mn - 5$
* Group $m$ terms: $5m - 4m + 2m = 3m$
* Group $n$ terms: $-7n + 3n = -4n$
* Group $mn$ terms: $-3mn$ (no other like terms)
* Group numbers: $2 - 5 = -3$
* Answer: $3m - 4n - 3mn - 3$
6. $4y(3y^2 + 5y - 7)$ and $2(y^3 - 4y^2 + 5)$
* First, expand the brackets by multiplying:
* Part 1: $12y^3 + 20y^2 - 28y$
* Part 2: $2y^3 - 8y^2 + 10$
* Now add them:
* $y^3$ terms: $12y^3 + 2y^3 = 14y^3$
* $y^2$ terms: $20y^2 - 8y^2 = 12y^2$
* $y$ terms: $-28y$
* Numbers: $+10$
* Answer: $14y^3 + 12y^2 - 28y + 10$
7. $5m(3 - m)$ and $6m^2 - 13m$
* Expand the first part: $15m - 5m^2$
* Add to the second part: $(15m - 5m^2) + (6m^2 - 13m)$
* Group $m^2$ terms: $-5m^2 + 6m^2 = m^2$
* Group $m$ terms: $15m - 13m = 2m$
* Answer: $m^2 + 2m$
8. $7xy + 5yz - 3zx$, $4yz + 9zx - 4y$, $-3xz + 5x - 2xy$
* Group $xy$ terms: $7xy - 2xy = 5xy$
* Group $yz$ terms: $5yz + 4yz = 9yz$
* Group $zx$ (or $xz$) terms: $-3zx + 9zx - 3xz = 3zx$
* Other terms: $-4y$ and $+5x$ stay as they are.
* Answer: $5xy + 9yz + 3zx - 4y + 5x$
---
1. $(a + b)(2a - 3b + c) - (2a - 3b)c$
* Multiply the first bracket: $a(2a - 3b + c) + b(2a - 3b + c) = 2a^2 - 3ab + ac + 2ab - 3b^2 + bc$
* Multiply the second part: $-(2ac - 3bc) = -2ac + 3bc$
* Combine everything: $2a^2 - 3ab + ac + 2ab - 3b^2 + bc - 2ac + 3bc$
* Simplify like terms:
* $a^2$: $2a^2$
* $ab$: $-3ab + 2ab = -ab$
* $ac$: $ac - 2ac = -ac$
* $b^2$: $-3b^2$
* $bc$: $bc + 3bc = 4bc$
* Answer: $2a^2 - ab - ac - 3b^2 + 4bc$
2. $(x + y)(2x + y) + (x + 2y)(x - y)$
* Expand first part: $2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2$
* Expand second part: $x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2$
* Add them together: $(2x^2 + x^2) + (3xy + xy) + (y^2 - 2y^2)$
* Answer: $3x^2 + 4xy - y^2$
3. $(a + b + c)(a + b - c)$
* This is in the form $(A+B)(A-B)$ where $A = (a+b)$ and $B = c$. The identity is $A^2 - B^2$.
* So, $(a + b)^2 - c^2$
* Expand $(a+b)^2$: $a^2 + 2ab + b^2$
* Subtract $c^2$: $a^2 + 2ab + b^2 - c^2$
* Answer: $a^2 + b^2 - c^2 + 2ab$
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Subtract $5x^2 - 4y^2 + 6y - 3$ from $7x^2 - 4xy + 8y^2 + 5x - 3y$
* Write it as: (Second Expression) - (First Expression)
* $(7x^2 - 4xy + 8y^2 + 5x - 3y) - (5x^2 - 4y^2 + 6y - 3)$
* Change signs of the second part: $-5x^2 + 4y^2 - 6y + 3$
* Combine like terms:
* $x^2$: $7x^2 - 5x^2 = 2x^2$
* $xy$: $-4xy$ (only one term)
* $y^2$: $8y^2 + 4y^2 = 12y^2$
* $x$: $5x$ (only one term)
* $y$: $-3y - 6y = -9y$
* Numbers: $+3$
* Answer: $2x^2 - 4xy + 12y^2 + 5x - 9y + 3$
---
Subtract $4a - 7ab + 3b + 12$ from $12a - 9ab + 5b - 3$
* $(12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12)$
* Change signs: $-4a + 7ab - 3b - 12$
* Combine like terms:
* $a$: $12a - 4a = 8a$
* $ab$: $-9ab + 7ab = -2ab$
* $b$: $5b - 3b = 2b$
* Numbers: $-3 - 12 = -15$
* Answer: $8a - 2ab + 2b - 15$
---
Subtract $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$
* $(5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx)$
* Change signs: $-3xy - 5yz + 7zx$
* Combine like terms:
* $xy$: $5xy - 3xy = 2xy$
* $yz$: $-2yz - 5yz = -7yz$
* $zx$: $-2zx + 7zx = 5zx$
* $xyz$: $10xyz$ (only one term)
* Answer: $2xy - 7yz + 5zx + 10xyz$
---
Subtract $24ab - 10b - 18a$ from $30ab + 12b + 14a$
* $(30ab + 12b + 14a) - (24ab - 10b - 18a)$
* Change signs: $-24ab + 10b + 18a$
* Combine like terms:
* $ab$: $30ab - 24ab = 6ab$
* $b$: $12b + 10b = 22b$
* $a$: $14a + 18a = 32a$
* Answer: $6ab + 22b + 32a$
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Final Answer:
Q.1
1. $-5tz$
2. $26mn - 4$
3. $a + b + 3$
4. $7x + 5$
5. $3m - 4n - 3mn - 3$
6. $14y^3 + 12y^2 - 28y + 10$
7. $m^2 + 2m$
8. $5xy + 9yz + 3zx - 4y + 5x$
Q.2
1. $2a^2 - ab - ac - 3b^2 + 4bc$
2. $3x^2 + 4xy - y^2$
3. $a^2 + b^2 - c^2 + 2ab$
Q.3
$2x^2 - 4xy + 12y^2 + 5x - 9y + 3$
Q.4
$8a - 2ab + 2b - 15$
Q.5
$2xy - 7yz + 5zx + 10xyz$
Q.6
$6ab + 22b + 32a$
Q.1 Add
To add algebraic expressions, we group "like terms" together (terms with the same variables and powers) and then add their numbers (coefficients).
1. $t - 8tz$, $3tz - z$, $z - t$
* Group like terms: $(t - t) + (-8tz + 3tz) + (-z + z)$
* Simplify: $0 - 5tz + 0$
* Answer: $-5tz$
2. $7mn + 5$, $12mn + 2$, $9mn - 8$, $-2mn - 3$
* Group $mn$ terms: $7mn + 12mn + 9mn - 2mn = 26mn$
* Group numbers: $5 + 2 - 8 - 3 = 7 - 11 = -4$
* Answer: $26mn - 4$
3. $a + b - 3$, $b - a + 3$, $a - b + 3$
* Group $a$ terms: $a - a + a = a$
* Group $b$ terms: $b + b - b = b$
* Group numbers: $-3 + 3 + 3 = 3$
* Answer: $a + b + 3$
4. $14x + 10y - 12xy - 13$, $18 - 7x - 10y + 8xy$, $4xy$
* Group $x$ terms: $14x - 7x = 7x$
* Group $y$ terms: $10y - 10y = 0$
* Group $xy$ terms: $-12xy + 8xy + 4xy = 0$
* Group numbers: $-13 + 18 = 5$
* Answer: $7x + 5$
5. $5m - 7n$, $3n - 4m + 2$, $2m - 3mn - 5$
* Group $m$ terms: $5m - 4m + 2m = 3m$
* Group $n$ terms: $-7n + 3n = -4n$
* Group $mn$ terms: $-3mn$ (no other like terms)
* Group numbers: $2 - 5 = -3$
* Answer: $3m - 4n - 3mn - 3$
6. $4y(3y^2 + 5y - 7)$ and $2(y^3 - 4y^2 + 5)$
* First, expand the brackets by multiplying:
* Part 1: $12y^3 + 20y^2 - 28y$
* Part 2: $2y^3 - 8y^2 + 10$
* Now add them:
* $y^3$ terms: $12y^3 + 2y^3 = 14y^3$
* $y^2$ terms: $20y^2 - 8y^2 = 12y^2$
* $y$ terms: $-28y$
* Numbers: $+10$
* Answer: $14y^3 + 12y^2 - 28y + 10$
7. $5m(3 - m)$ and $6m^2 - 13m$
* Expand the first part: $15m - 5m^2$
* Add to the second part: $(15m - 5m^2) + (6m^2 - 13m)$
* Group $m^2$ terms: $-5m^2 + 6m^2 = m^2$
* Group $m$ terms: $15m - 13m = 2m$
* Answer: $m^2 + 2m$
8. $7xy + 5yz - 3zx$, $4yz + 9zx - 4y$, $-3xz + 5x - 2xy$
* Group $xy$ terms: $7xy - 2xy = 5xy$
* Group $yz$ terms: $5yz + 4yz = 9yz$
* Group $zx$ (or $xz$) terms: $-3zx + 9zx - 3xz = 3zx$
* Other terms: $-4y$ and $+5x$ stay as they are.
* Answer: $5xy + 9yz + 3zx - 4y + 5x$
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Q.2 Simplify
1. $(a + b)(2a - 3b + c) - (2a - 3b)c$
* Multiply the first bracket: $a(2a - 3b + c) + b(2a - 3b + c) = 2a^2 - 3ab + ac + 2ab - 3b^2 + bc$
* Multiply the second part: $-(2ac - 3bc) = -2ac + 3bc$
* Combine everything: $2a^2 - 3ab + ac + 2ab - 3b^2 + bc - 2ac + 3bc$
* Simplify like terms:
* $a^2$: $2a^2$
* $ab$: $-3ab + 2ab = -ab$
* $ac$: $ac - 2ac = -ac$
* $b^2$: $-3b^2$
* $bc$: $bc + 3bc = 4bc$
* Answer: $2a^2 - ab - ac - 3b^2 + 4bc$
2. $(x + y)(2x + y) + (x + 2y)(x - y)$
* Expand first part: $2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2$
* Expand second part: $x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2$
* Add them together: $(2x^2 + x^2) + (3xy + xy) + (y^2 - 2y^2)$
* Answer: $3x^2 + 4xy - y^2$
3. $(a + b + c)(a + b - c)$
* This is in the form $(A+B)(A-B)$ where $A = (a+b)$ and $B = c$. The identity is $A^2 - B^2$.
* So, $(a + b)^2 - c^2$
* Expand $(a+b)^2$: $a^2 + 2ab + b^2$
* Subtract $c^2$: $a^2 + 2ab + b^2 - c^2$
* Answer: $a^2 + b^2 - c^2 + 2ab$
---
Q.3 Subtract
Subtract $5x^2 - 4y^2 + 6y - 3$ from $7x^2 - 4xy + 8y^2 + 5x - 3y$
* Write it as: (Second Expression) - (First Expression)
* $(7x^2 - 4xy + 8y^2 + 5x - 3y) - (5x^2 - 4y^2 + 6y - 3)$
* Change signs of the second part: $-5x^2 + 4y^2 - 6y + 3$
* Combine like terms:
* $x^2$: $7x^2 - 5x^2 = 2x^2$
* $xy$: $-4xy$ (only one term)
* $y^2$: $8y^2 + 4y^2 = 12y^2$
* $x$: $5x$ (only one term)
* $y$: $-3y - 6y = -9y$
* Numbers: $+3$
* Answer: $2x^2 - 4xy + 12y^2 + 5x - 9y + 3$
---
Q.4 Subtract
Subtract $4a - 7ab + 3b + 12$ from $12a - 9ab + 5b - 3$
* $(12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12)$
* Change signs: $-4a + 7ab - 3b - 12$
* Combine like terms:
* $a$: $12a - 4a = 8a$
* $ab$: $-9ab + 7ab = -2ab$
* $b$: $5b - 3b = 2b$
* Numbers: $-3 - 12 = -15$
* Answer: $8a - 2ab + 2b - 15$
---
Q.5 Subtract
Subtract $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$
* $(5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx)$
* Change signs: $-3xy - 5yz + 7zx$
* Combine like terms:
* $xy$: $5xy - 3xy = 2xy$
* $yz$: $-2yz - 5yz = -7yz$
* $zx$: $-2zx + 7zx = 5zx$
* $xyz$: $10xyz$ (only one term)
* Answer: $2xy - 7yz + 5zx + 10xyz$
---
Q.6 Subtract
Subtract $24ab - 10b - 18a$ from $30ab + 12b + 14a$
* $(30ab + 12b + 14a) - (24ab - 10b - 18a)$
* Change signs: $-24ab + 10b + 18a$
* Combine like terms:
* $ab$: $30ab - 24ab = 6ab$
* $b$: $12b + 10b = 22b$
* $a$: $14a + 18a = 32a$
* Answer: $6ab + 22b + 32a$
──────────────────────────────────────
Final Answer:
Q.1
1. $-5tz$
2. $26mn - 4$
3. $a + b + 3$
4. $7x + 5$
5. $3m - 4n - 3mn - 3$
6. $14y^3 + 12y^2 - 28y + 10$
7. $m^2 + 2m$
8. $5xy + 9yz + 3zx - 4y + 5x$
Q.2
1. $2a^2 - ab - ac - 3b^2 + 4bc$
2. $3x^2 + 4xy - y^2$
3. $a^2 + b^2 - c^2 + 2ab$
Q.3
$2x^2 - 4xy + 12y^2 + 5x - 9y + 3$
Q.4
$8a - 2ab + 2b - 15$
Q.5
$2xy - 7yz + 5zx + 10xyz$
Q.6
$6ab + 22b + 32a$
Parent Tip: Review the logic above to help your child master the concept of algebraic expressions worksheets.
