Problem A: 6a + (8b + 2a)
e1. = 6a + (2a + 8b)
Asymmetric
Bdistributive
Cmultiplicative identity
Dcommutative
Ereflexive
Fsubstitution
f2. = (6a + 2a) +8b
Adistributive
Bassociative
Csubstitution
Dmultiplicative property of zero
Emultiplicative identity
Fadditive identity
b3. = (6 + 2)a + 8b
Areflexive
Bassociative
Csubstitution
Dcommutative
Etransitive
Fdistributive
c4. = 8a + 8b
Aassociative
Breflexive
Cadditive identity
Dcommutative
Esubstitution
Fsymmetric
Problem B: 8a^2 + (8a+a^2) + 7a
c5. = 8a^2 + (a^2 + 8a) + 7a
Acommutative
Btransitive
Cassociative
Dreflexive
Esymmetric
Fmultiplicative property of zero
b6. = (8a^2 + a^2) + 8a + 7a
Asymmetric
Bcommutative
Creflexive
Ddistributive
Emultiplicative identity
Fassociative
For numbers 7 – 12, Combine like terms
c7. 9y^2 + 13y^2 + 3
A22y^2 + 3
By
C 22y^4
D25y
E25y^2
F22y
a8. 12a^2 + 14a - 11a^2
A37a^5
Ba^2 + 14a
C22a^3
D11a^2
E11a
F33a^3
d9. 14a^2 + 27 + 13a^2
A54a
B27a^2
C54a^2
D27a^2 + 27
E27(a + 1)
F27a^5
d10. 5a + 10b + 7 + 5b
A5a + 15b^2 + 7
B20ab + 7
C15ab + 12b
D5a + 15b + 7
E10ab^2 + 7
F27ab^2
c11. 3(x + 2y) - 2y
A3x
B3x + 8y
C3x + 4y
Dx^3 + 6y^3
Ex^3
F3x + y
b12. 5x + 3(x - y)
A5x^2 + 3x - 3y
B8x - 3y
C15x^2 - 3y
D2x - 3y
E2x - y
F5x^4 - y^4
Write the phrase as a mathematical expression. Use x to represent the number. Combine like terms if possible.
f13. Seven times a number, added to the sum of the number and eight.
A7 + n + 8
B7 – n + 8
C56n
D7 + 8n
E7n + (n + 8)
F(7n) + (8n)
a14. A number multiplied by -7, subtracted from the sum of 11 and three times the number
A-7n – (11 + 3n)
B(-7 – 11) -3 n
C11 - 7n
D(11 + 3n) - (-7n)
E3n(7 - -11)
F(11 – 3n) - (-7n)
e15. A number multiplied by 9, added to -8, subtracted from 5 times the number
A9n + 8 - 5(5n)
B5n – 9+ 8
C9n + 8 – 5n
D 5n – 9 + (-8)
E5n – [9n + (-8)]
F5n – 9n + 8
a16. 2 times a number added to -7, added to 6 times the sum of 3 times the number and 8
A2n + (-7) + 6(3n + 8)
B2n + 7 – 6 + 3n +8
C2 + (-7) + 6 + 3n (8)
D2n + (-7) + 6 + 3n + 8
E(6)3n + 8(2n) + (-7)
F-7 = 6(2n + 3n) + 8
e17. Seven times a number added to 7 subtracted from triple the sum of six times the number and 7
A7n + 3+ 6n - 7
B7n + 6n + 7(3)
C6n + 7 – 7n + 7
D(7n + 7) – 3(6n + 7)
E7n + 7 – 3(6n + 7)
F3(6n + 7) – (7n + 7)
Problem A:
1. 6a + (8b + 2a)
= 6a + 2a + 8b (using the associative property)
= 8a + 8b (combining like terms)
Answer: 8a + 8b
Problem B:
1. 8a^2 + (8a + a^2) + 7a
= 8a^2 + a^2 + 8a + 7a (using the associative property)
= 9a^2 + 15a (combining like terms)
Answer: 9a^2 + 15a
For numbers 7 - 12, combine like terms:
7. 9y^2 + 13y^2 + 3
= 22y^2 + 3 (combining like terms)
Answer: 22y^2 + 3
8. 12a^2 + 14a - 11a^2
= (12a^2 - 11a^2) + 14a (combining like terms)
= a^2 + 14a (simplified)
Answer: a^2 + 14a
9. 14a^2 + 27 + 13a^2
= (14a^2 + 13a^2) + 27 (combining like terms)
= 27a^2 + 27 (simplified)
Answer: 27a^2 + 27
10. 5a + 10b + 7 + 5b
= 5a + 5b + 10b + 7 (rearranging terms)
= 5a + 15b + 7 (combining like terms)
Answer: 5a + 15b + 7
11. 3(x + 2y) - 2y
= 3x + 6y - 2y (distributing and combining like terms)
= 3x + 4y (simplified)
Answer: 3x + 4y
12. 5x + 3(x - y)
= 5x + 3x - 3y (distributing and combining like terms)
= 8x - 3y (simplified)
Answer: 8x - 3y
Write the phrase as a mathematical expression:
13. Seven times a number, added to the sum of the number and eight.
= 7n + (n + 8) (rewriting the phrase as an expression)
Answer: 7n + n + 8
14. A number multiplied by -7, subtracted from the sum of 11 and three times the number.
= (11 + 3n) - (-7n) (rearranging terms and distributing)
= 11 + 3n + 7n (combining like terms)
Answer: 11 + 10n
15. A number multiplied by 9, added to -8, subtracted from 5 times the number.
= 9n - 8 + (5n - 9) (rearranging terms and distributing)
= 14n - 17 (combining like terms)
Answer: 14n - 17
16. 2 times a number added to -7, added to 6 times the sum of 3 times the number and 8.
= 2n - 7 + 6(3n + 8) (distributing and combining like terms)
= 2n - 7 + 18n + 48 (simplified)
Answer: 20n + 41
17. Seven times a number added to 7, subtracted from triple the sum of six times the number and 7.
= 3(6n + 7) - (7n + 7) (distributing and rearranging terms)
= 18n + 21 - 7n - 7 (combining like terms)
Answer: 11n + 14
To solve these problems, we need to apply the rules and properties of algebraic expressions. Let's go through each problem step by step:
Problem A: 6a + (8b + 2a)
e1. = 6a + (2a + 8b)
Rearranging the terms inside the parentheses using the commutative property, we get:
= 6a + (8b + 2a)
f2. = (6a + 2a) + 8b
Using the associative property to group like terms together, we get:
= (6a + 2a) + 8b
b3. = (6 + 2)a + 8b
Combining the coefficients of like terms, we get:
= 8a + 8b
c4. = 8a + 8b
Combining the coefficients of like terms, we get:
= 8a + 8b
So, the simplified form of the expression 6a + (8b + 2a) is 8a + 8b.
Problem B: 8a^2 + (8a + a^2) + 7a
c5. = 8a^2 + (a^2 + 8a) + 7a
Rearranging the terms inside the parentheses using the commutative property, we get:
= 8a^2 + (8a + a^2) + 7a
b6. = (8a^2 + a^2) + 8a + 7a
Using the associative property to group like terms together, we get:
= (8a^2 + a^2) + 8a + 7a
So, the simplified form of the expression 8a^2 + (8a + a^2) + 7a is (8a^2 + a^2) + 8a + 7a.
For the remaining problems, we need to combine like terms:
c7. 9y^2 + 13y^2 + 3
Combining the coefficients of like terms, we get:
= (9y^2 + 13y^2) + 3
= 22y^2 + 3
a8. 12a^2 + 14a - 11a^2
Combining the coefficients of like terms, we get:
= (12a^2 - 11a^2) + 14a
= a^2 + 14a
d9. 14a^2 + 27 + 13a^2
Combining the coefficients of like terms, we get:
= (14a^2 + 13a^2) + 27
= 27a^2 + 27
d10. 5a + 10b + 7 + 5b
Combining the coefficients of like terms, we get:
= 5a + (10b + 5b) + 7
= 5a + 15b + 7
c11. 3(x + 2y) - 2y
Distributing the multiplication, we get:
= 3x + 6y - 2y
Combining the coefficients of like terms, we get:
= 3x + 4y
b12. 5x + 3(x - y)
Distributing the multiplication, we get:
= 5x + 3x - 3y
Combining the coefficients of like terms, we get:
= 8x - 3y
f13. Seven times a number, added to the sum of the number and eight.
The mathematical expression is: 7n + (n + 8)
a14. A number multiplied by -7, subtracted from the sum of 11 and three times the number.
The mathematical expression is: (11 + 3n) - (-7n)
e15. A number multiplied by 9, added to -8, subtracted from 5 times the number.
The mathematical expression is: 5n - (9n + 8)
a16. 2 times a number added to -7, added to 6 times the sum of 3 times the number and 8.
The mathematical expression is: 2n + (-7) + 6(3n + 8)
e17. Seven times a number added to 7, subtracted from triple the sum of six times the number and 7.
The mathematical expression is: 3(6n + 7) - (7n + 7)
I hope this helps! Let me know if you have any further questions.