c. $14x + 2$ and $2(7x + 1)$ $x = 10$
d. explain any patterns that you notice in the results to parts (a)-(c).
e. what would happen if other values were given for the variables?
closing
how can you use your knowledge of gcf and the distributive property to write equivalent expressions?
find the missing value that makes the two expressions equivalent.
$4x + 12y$ ______$(x + 3y)$
$35x + 50y$ ______$(7x + 10y)$
$18x + 9y$ ______$(2x + y)$
$32x + 8y$ ______$(4x + y)$
$100x + 700y$ ______$(x + 7y)$
explain how you determine the missing number.

Question

Accepted Answer

### First, solving the missing value problems (Step-by-Step Format)
# Explanation:
## Step1: Find GCF of 4 and 12
GCF(4,12) = 4, so $\frac{4x+12y}{x+3y}=4$
## Step2: Find GCF of 35 and 50
GCF(35,50) = 5, so $\frac{35x+50y}{7x+10y}=5$
## Step3: Find GCF of 18 and 9
GCF(18,9) = 9, so $\frac{18x+9y}{2x+y}=9$
## Step4: Find GCF of 32 and 8
GCF(32,8) = 8, so $\frac{32x+8y}{4x+y}=8$
## Step5: Find GCF of 100 and 700
GCF(100,700) = 100, so $\frac{100x+700y}{x+7y}=100$

# Answer:
1. $4$
2. $5$
3. $9$
4. $8$
5. $100$

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### Explanation for determining the missing number (Answer-Explanation Format)
# Brief Explanations:
The missing number is the greatest common factor (GCF) of the coefficients of the terms in the original expression. Use the distributive property in reverse: factor out the GCF from each term of the left-hand expression, which leaves the expression in the parentheses. Divide each coefficient in the original expression by the corresponding coefficient in the parenthetical expression to find this GCF.
# Answer:
The missing number is the GCF of the coefficients of the terms in the left-hand expression. Calculate the GCF by identifying the largest number that divides evenly into all coefficients of the original expression, or divide each coefficient in the original expression by the matching coefficient in the parenthetical expression to find the factor.

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### Addressing parts (d), (e), and the closing question (Answer-Explanation Format)
# Brief Explanations:
- For (d): In parts (a)-(c), the paired expressions are equivalent (they simplify to the same value for the given $x$), because one is the factored form of the other using the distributive property.
- For (e): If other values are used for the variables, the paired expressions will still be equal, since they are algebraically equivalent (the distributive property holds for all real numbers).
- Closing question: To write equivalent expressions, first find the GCF of all terms in the expression. Then use the reverse distributive property (factoring): divide each term by the GCF, place the GCF outside parentheses, and the resulting quotient terms inside parentheses. To expand a factored expression, use the distributive property: multiply the factor outside the parentheses by each term inside.
# Answer:
- (d): The paired expressions in (a)-(c) are equivalent (they have the same value for $x=10$) because one is the factored form of the other.
- (e): The paired expressions will still have equal values for any variable input, as they are algebraically equivalent.
- Closing: To write equivalent expressions, factor out the GCF of all terms using the reverse distributive property (place the GCF outside parentheses and the simplified terms inside), or expand a factored expression by multiplying the outside factor by each term inside via the distributive property.

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QUESTION IMAGE

Question

Explanation:

First, solving the missing value problems (Step-by-Step Format)

Step1: Find GCF of 4 and 12

Step2: Find GCF of 35 and 50

Step3: Find GCF of 18 and 9

Step4: Find GCF of 32 and 8

Step5: Find GCF of 100 and 700

Answer: