Question

1. rewrite each of the following in simplest form by combining like terms and constants (normal numbers with no variables attached). rewrite subtraction as addition when necessary.

(a) $5x + 7 + 6x + 2$
(b) $7y + 4 - 2y + 10$
(c) $6c - 2 - 9c - 8$
(d) $11x + 2y + 4x + 8y$
(e) $3m + 2n - 5m + 11n$
(f) $-2w + 4z + 7w - 9z$
(g) $15x + 2y + 7 + 6y + 4x + 2$
(h) $3a + 7b - 9 + 8a - 3b + 4$

1. consider the expression $8y + 7 + 2y + 11$.

(a) for step #1 and step #2, state the property that can be used to justify it.
original: $8y + 7 + 2y + 11$
step #1: $8y + 2y + 7 + 11$ property: ____________
step #2: $(8y + 2y)+(7 + 11)$ property: ____________
step #3: $10y + 18$
(b) test to see if the two expressions are equivalent by substituting $y = 3$ into both. show your substitutions and steps in evaluating each expressions value.
$8y + 7 + 2y + 11$
$10y + 18$

Explanation:

Problem 6

Step1: Group like terms

Step2: Combine like terms

---

(a)

Step1: Group x-terms and constants

$5x + 6x + 7 + 2$

Step2: Combine like terms

$(5+6)x + (7+2) = 11x + 9$
---

(b)

Step1: Group y-terms and constants

$7y - 2y + 4 + 10$

Step2: Combine like terms

$(7-2)y + (4+10) = 5y + 14$
---

(c)

Step1: Group c-terms and constants

$6c - 9c - 2 - 8$

Step2: Combine like terms

$(6-9)c + (-2-8) = -3c - 10$
---

(d)

Step1: Group x-terms and y-terms

$11x + 4x + 2y + 8y$

Step2: Combine like terms

$(11+4)x + (2+8)y = 15x + 10y$
---

(e)

Step1: Group m-terms and n-terms

$3m - 5m + 2n + 11n$

Step2: Combine like terms

$(3-5)m + (2+11)n = -2m + 13n$
---

(f)

Step1: Group w-terms and z-terms

$-2w + 7w + 4z - 9z$

Step2: Combine like terms

$(-2+7)w + (4-9)z = 5w - 5z$
---

(g)

Step1: Group x, y terms, constants

$15x + 4x + 2y + 6y + 7 + 2$

Step2: Combine like terms

$(15+4)x + (2+6)y + (7+2) = 19x + 8y + 9$
---

(h)

Step1: Group a, b terms, constants

$3a + 8a + 7b - 3b - 9 + 4$

Step2: Combine like terms

$(3+8)a + (7-3)b + (-9+4) = 11a + 4b - 5$

Step #1 rearranges the order of terms without changing their value, which is defined by the Commutative Property of Addition. Step #2 groups like terms together to simplify, which is defined by the Associative Property of Addition.

Step1: Substitute $y=3$ into first expression

$8(3) + 7 + 2(3) + 11$

Step2: Calculate the first expression

$24 + 7 + 6 + 11 = 48$

Step3: Substitute $y=3$ into second expression

$10(3) + 18$

Step4: Calculate the second expression

$30 + 18 = 48$

Answer:

(a) $11x + 9$
(b) $5y + 14$
(c) $-3c - 10$
(d) $15x + 10y$
(e) $-2m + 13n$
(f) $5w - 5z$
(g) $19x + 8y + 9$
(h) $11a + 4b - 5$

---

Problem 7

(a)