Answer :
Sure! Let's walk through these problems step by step to simplify them where possible.
### Given Expressions:
#### 1. No Simplification Required:
e) [tex]\(3a - 4b + 6c\)[/tex]
- This expression cannot be simplified further because it contains different variables.
f) [tex]\(41bc + 42bd - 43ac\)[/tex]
- Similarly, this expression also contains different variables and thus cannot be simplified further.
#### 2. Simplify if Possible:
a) [tex]\(26a + 4a - 8a\)[/tex]
- Combine the like terms:
- [tex]\(26a + 4a = 30a\)[/tex]
- [tex]\(30a - 8a = 22a\)[/tex]
- Therefore, the simplified form is [tex]\(22a\)[/tex].
c) [tex]\(3a - 2a^2\)[/tex]
- This expression contains different powers of [tex]\(a\)[/tex] and cannot be simplified further.
b) [tex]\(17x - 12x + 33x\)[/tex]
- Combine the like terms:
- [tex]\(17x - 12x = 5x\)[/tex]
- [tex]\(5x + 33x = 38x\)[/tex]
- Therefore, the simplified form is [tex]\(38x\)[/tex].
e) [tex]\(y + 2y - 3y\)[/tex]
- Combine the like terms:
- [tex]\(y + 2y = 3y\)[/tex]
- [tex]\(3y - 3y = 0y\)[/tex], which is simply [tex]\(0\)[/tex].
- Therefore, the simplified form is [tex]\(0\)[/tex].
d) [tex]\(15abc - 22bca + 13\)[/tex]
- Notice that [tex]\(15abc\)[/tex] and [tex]\(-22bca\)[/tex] are like terms (since the variables are the same and multiplication is commutative):
- [tex]\(15abc - 22abc = -7abc\)[/tex]
- Therefore, the simplified form is [tex]\(-7abc + 13\)[/tex].
f) [tex]\(10m^2n - 6m^2n + 4m^2n\)[/tex]
- Combine the like terms:
- [tex]\(10m^2n - 6m^2n = 4m^2n\)[/tex]
- [tex]\(4m^2n + 4m^2n = 8m^2n\)[/tex]
- Therefore, the simplified form is [tex]\(8m^2n\)[/tex].
#### 3. Simplify if Possible:
a) [tex]\(2a - 3b + 4a - 5b\)[/tex]
- Combine the like terms separately:
- For [tex]\(a\)[/tex]:
- [tex]\(2a + 4a = 6a\)[/tex]
- For [tex]\(b\)[/tex]:
- [tex]\(-3b - 5b = -8b\)[/tex]
- Therefore, the simplified form is [tex]\(6a - 8b\)[/tex].
b) [tex]\(6xy + 2xy - xy\)[/tex]
- Combine the like terms:
- [tex]\(6xy + 2xy = 8xy\)[/tex]
- [tex]\(8xy - xy = 7xy\)[/tex]
- Therefore, the simplified form is [tex]\(7xy\)[/tex].
c) [tex]\(n + n + n + n\)[/tex]
- Combine the like terms:
- [tex]\(n + n + n + n = 4n\)[/tex]
- Therefore, the simplified form is [tex]\(4n\)[/tex].
d) [tex]\(n + m + n + m\)[/tex]
- Combine the like terms separately:
- For [tex]\(n\)[/tex]:
- [tex]\(n + n = 2n\)[/tex]
- For [tex]\(m\)[/tex]:
- [tex]\(m + m = 2m\)[/tex]
- Therefore, the simplified form is [tex]\(2n + 2m\)[/tex].
e) [tex]\(9s - s\)[/tex]
- Combine the like terms:
- [tex]\(9s - s = 8s\)[/tex]
- Therefore, the simplified form is [tex]\(8s\)[/tex].
f) [tex]\(7cd + 4df + 3cd - 4df\)[/tex]
- Combine the like terms separately:
- For [tex]\(cd\)[/tex]:
- [tex]\(7cd + 3cd = 10cd\)[/tex]
- For [tex]\(df\)[/tex]:
- [tex]\(4df - 4df = 0df\)[/tex], which can be omitted
- Therefore, the simplified form is [tex]\(10cd\)[/tex].
### Summary of Solutions:
1. e) [tex]\(3a - 4b + 6c\)[/tex]
2. f) [tex]\(41bc + 42bd - 43ac\)[/tex]
3. a) [tex]\(22a\)[/tex]
4. c) [tex]\(3a - 2a^2\)[/tex]
5. b) [tex]\(38x\)[/tex]
6. e) [tex]\(0\)[/tex]
7. d) [tex]\(-7abc + 13\)[/tex]
8. f) [tex]\(8m^2n\)[/tex]
9. a) [tex]\(6a - 8b\)[/tex]
10. b) [tex]\(7xy\)[/tex]
11. c) [tex]\(4n\)[/tex]
12. d) [tex]\(2n + 2m\)[/tex]
13. e) [tex]\(8s\)[/tex]
14. f) [tex]\(10cd\)[/tex]
### Given Expressions:
#### 1. No Simplification Required:
e) [tex]\(3a - 4b + 6c\)[/tex]
- This expression cannot be simplified further because it contains different variables.
f) [tex]\(41bc + 42bd - 43ac\)[/tex]
- Similarly, this expression also contains different variables and thus cannot be simplified further.
#### 2. Simplify if Possible:
a) [tex]\(26a + 4a - 8a\)[/tex]
- Combine the like terms:
- [tex]\(26a + 4a = 30a\)[/tex]
- [tex]\(30a - 8a = 22a\)[/tex]
- Therefore, the simplified form is [tex]\(22a\)[/tex].
c) [tex]\(3a - 2a^2\)[/tex]
- This expression contains different powers of [tex]\(a\)[/tex] and cannot be simplified further.
b) [tex]\(17x - 12x + 33x\)[/tex]
- Combine the like terms:
- [tex]\(17x - 12x = 5x\)[/tex]
- [tex]\(5x + 33x = 38x\)[/tex]
- Therefore, the simplified form is [tex]\(38x\)[/tex].
e) [tex]\(y + 2y - 3y\)[/tex]
- Combine the like terms:
- [tex]\(y + 2y = 3y\)[/tex]
- [tex]\(3y - 3y = 0y\)[/tex], which is simply [tex]\(0\)[/tex].
- Therefore, the simplified form is [tex]\(0\)[/tex].
d) [tex]\(15abc - 22bca + 13\)[/tex]
- Notice that [tex]\(15abc\)[/tex] and [tex]\(-22bca\)[/tex] are like terms (since the variables are the same and multiplication is commutative):
- [tex]\(15abc - 22abc = -7abc\)[/tex]
- Therefore, the simplified form is [tex]\(-7abc + 13\)[/tex].
f) [tex]\(10m^2n - 6m^2n + 4m^2n\)[/tex]
- Combine the like terms:
- [tex]\(10m^2n - 6m^2n = 4m^2n\)[/tex]
- [tex]\(4m^2n + 4m^2n = 8m^2n\)[/tex]
- Therefore, the simplified form is [tex]\(8m^2n\)[/tex].
#### 3. Simplify if Possible:
a) [tex]\(2a - 3b + 4a - 5b\)[/tex]
- Combine the like terms separately:
- For [tex]\(a\)[/tex]:
- [tex]\(2a + 4a = 6a\)[/tex]
- For [tex]\(b\)[/tex]:
- [tex]\(-3b - 5b = -8b\)[/tex]
- Therefore, the simplified form is [tex]\(6a - 8b\)[/tex].
b) [tex]\(6xy + 2xy - xy\)[/tex]
- Combine the like terms:
- [tex]\(6xy + 2xy = 8xy\)[/tex]
- [tex]\(8xy - xy = 7xy\)[/tex]
- Therefore, the simplified form is [tex]\(7xy\)[/tex].
c) [tex]\(n + n + n + n\)[/tex]
- Combine the like terms:
- [tex]\(n + n + n + n = 4n\)[/tex]
- Therefore, the simplified form is [tex]\(4n\)[/tex].
d) [tex]\(n + m + n + m\)[/tex]
- Combine the like terms separately:
- For [tex]\(n\)[/tex]:
- [tex]\(n + n = 2n\)[/tex]
- For [tex]\(m\)[/tex]:
- [tex]\(m + m = 2m\)[/tex]
- Therefore, the simplified form is [tex]\(2n + 2m\)[/tex].
e) [tex]\(9s - s\)[/tex]
- Combine the like terms:
- [tex]\(9s - s = 8s\)[/tex]
- Therefore, the simplified form is [tex]\(8s\)[/tex].
f) [tex]\(7cd + 4df + 3cd - 4df\)[/tex]
- Combine the like terms separately:
- For [tex]\(cd\)[/tex]:
- [tex]\(7cd + 3cd = 10cd\)[/tex]
- For [tex]\(df\)[/tex]:
- [tex]\(4df - 4df = 0df\)[/tex], which can be omitted
- Therefore, the simplified form is [tex]\(10cd\)[/tex].
### Summary of Solutions:
1. e) [tex]\(3a - 4b + 6c\)[/tex]
2. f) [tex]\(41bc + 42bd - 43ac\)[/tex]
3. a) [tex]\(22a\)[/tex]
4. c) [tex]\(3a - 2a^2\)[/tex]
5. b) [tex]\(38x\)[/tex]
6. e) [tex]\(0\)[/tex]
7. d) [tex]\(-7abc + 13\)[/tex]
8. f) [tex]\(8m^2n\)[/tex]
9. a) [tex]\(6a - 8b\)[/tex]
10. b) [tex]\(7xy\)[/tex]
11. c) [tex]\(4n\)[/tex]
12. d) [tex]\(2n + 2m\)[/tex]
13. e) [tex]\(8s\)[/tex]
14. f) [tex]\(10cd\)[/tex]
