name
practice & problem solving
for 7–9, write an equivalent expression.
7. -3(7 + 5g)
8. (x + 7) + 3y
9. $\frac{2}{9} - \frac{1}{3} + x$
10. which expression is equivalent to t + 4 + 3 - 2t?
a t + 7
b -t + 7
c 6t
d 10t
11. the distance in feet that karina swims in a race
is represented by 4d - 4, where d is the distance
for each lap. what is an expression equivalent
to 4d - 4?
(there is an image of a swimming pool with swimmers, one labeled karina, and a length labeled d)
12. use the associative property to write an expression equivalent
to (w + 9) + 3.
13. nigel is planning his training schedule for a marathon over a 4-day
period. he is uncertain how many miles he will run on two days.
one expression for the total miles he will run is 12 + y + 17 + z.
use the commutative property to write an equivalent expression.
(there is a chart labeled marathon training plan with columns day and miles to run, rows: 1 - 12, 2 - y, 3 - 17, 4 - z)
4-2 generate equivalent expressions 199
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Question

Accepted Answer

Let's solve each problem one by one:

### Problem 7:
# Explanation:
## Step 1: Apply the distributive property
The distributive property states that $ a(b + c) = ab + ac $. Here, $ a = -3 $, $ b = 7 $, and $ c = 5g $.
$ -3(7 + 5g) = -3 \times 7 + (-3) \times 5g $
## Step 2: Simplify the multiplication
$ -3 \times 7 = -21 $ and $ -3 \times 5g = -15g $
So, $ -3(7 + 5g) = -21 - 15g $
# Answer:
$ -21 - 15g $

### Problem 8:
# Explanation:
## Step 1: Apply the associative property of addition
The associative property of addition states that $ (a + b) + c = a + (b + c) $. Here, we can remove the parentheses and rewrite the expression as $ x + 7 + 3y $ (or we can also group $ x $ and $ 3y $ first, but the simplest equivalent is just removing the parentheses for addition).
Wait, actually, the original expression is $ (x + 7) + 3y $. Using the associative property, we can rewrite it as $ x + (7 + 3y) $ or just $ x + 7 + 3y $.
# Answer:
$ x + 7 + 3y $ (or $ x + (7 + 3y) $)

### Problem 9:
# Explanation:
## Step 1: Combine the like terms (the constant terms)
First, we have $ \frac{2}{9} - \frac{1}{3} + x $. Let's convert $ \frac{1}{3} $ to ninths: $ \frac{1}{3} = \frac{3}{9} $
## Step 2: Subtract the fractions
$ \frac{2}{9} - \frac{3}{9} = -\frac{1}{9} $
So the equivalent expression is $ -\frac{1}{9} + x $ (or $ x - \frac{1}{9} $)
# Answer:
$ x - \frac{1}{9} $ (or $ -\frac{1}{9} + x $)

### Problem 10:
# Explanation:
## Step 1: Combine like terms
The expression is $ t + 4 + 3 - 2t $. Combine the $ t $ terms: $ t - 2t = -t $
## Step 2: Combine the constant terms
$ 4 + 3 = 7 $
So the simplified expression is $ -t + 7 $
Now let's check the options:
A. $ t + 7 $ – Incorrect, we have $ -t $ not $ t $
B. $ -t + 7 $ – Correct
C. $ 6t $ – Incorrect, no multiplication of $ t $
D. $ 10t $ – Incorrect
# Answer:
B. $ -t + 7 $

### Problem 11:
# Explanation:
## Step 1: Factor out the common factor
The expression is $ 4d^2 - 4 $. The common factor of $ 4d^2 $ and $ 4 $ is $ 4 $.
Using the distributive property in reverse (factoring), we get $ 4(d^2 - 1) $
We can also factor $ d^2 - 1 $ as a difference of squares: $ d^2 - 1 = (d - 1)(d + 1) $, so another equivalent expression is $ 4(d - 1)(d + 1) $, but the simplest factoring is $ 4(d^2 - 1) $
# Answer:
$ 4(d^2 - 1) $ (or $ 4(d - 1)(d + 1) $)

### Problem 12:
# Explanation:
## Step 1: Recall the associative property of addition
The associative property of addition states that $ (a + b) + c = a + (b + c) $. Here, $ a = w $, $ b = 9 $, and $ c = 3 $.
So, $ (w + 9) + 3 = w + (9 + 3) $
## Step 2: Simplify the parentheses (optional, but the equivalent expression using the associative property is $ w + (9 + 3) $)
# Answer:
$ w + (9 + 3) $ (or $ w + 12 $ if we simplify $ 9 + 3 $)

### Problem 13:
# Explanation:
## Step 1: Recall the commutative property of addition
The commutative property of addition states that $ a + b = b + a $. The original expression is $ 12 + y + 17 + z $ (wait, the problem says "the total miles he will run is $ 12 + y + 17 + z $"? Wait, the user wrote: "One expression for the total miles he will run is $ 12 + y + 17 + z $. Use the Commutative Property to write an equivalent expression."
Wait, the original expression is $ 12 + y + 17 + z $. Using the commutative property, we can rearrange the terms. For example, we can write $ y + 12 + 17 + z $, or $ 12 + 17 + y + z $, etc. The simplest is to combine the constant terms first: $ 12 + 17 + y + z = 29 + y + z $, but using the commutative property, we can just rearrange the order of addition. Let's take the original expression $ 12 + y + 17 + z $. Using the commutative property, we can swap $ y $ with $ 17 $, or $ 12 $ with $ y $, etc. The most straightforward is to write $ y + 12 + 17 + z $ or $ 12 + 17 + y + z $. Let's do $ 12 + 17 + y + z $ first, then simplify $ 12 + 17 = 29 $, so $ 29 + y + z $ is also equivalent.
# Answer:
$ y + 12 + 17 + z $ (or $ 12 + 17 + y + z $ or $ 29 + y + z $)

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QUESTION IMAGE

Question

Explanation:

Problem 7:

Step 1: Apply the distributive property

Step 2: Simplify the multiplication

Step 1: Apply the associative property of addition

Step 1: Combine the like terms (the constant terms)

Step 2: Subtract the fractions

Answer:

Problem 8: