Question

name:
practice & problem solving
for 7–9, write an equivalent expression.

1. $-3(7 + 5g)$
2. $(x + 7) + 3y$
3. $\frac{2}{9} - \frac{1}{5} + x$
4. which expression is equivalent to $t + 4 + 3 - 2t$?

a $t + 7$
b $-t + 7$
c $6t$
d $10t$

1. 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 $4d - 4$?

image of swimmers with karina labeled

1. use the associative property to write an expression equivalent to $(w + 9) + 3$.
2. 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.

marathon training plan chart: day 1: 12, day 2: y, day 3: 17, day 4: z
4 - 2 generate equivalent expressions 199
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Explanation:

Let's solve each problem one by one:

Problem 7: $-3(7 + 5g)$

Step 1: Apply Distributive Property

The distributive property states that $a(b + c) = ab + ac$. Here, $a = -3$, $b = 7$, and $c = 5g$.
So, $-3(7) + (-3)(5g)$

Step 2: Simplify Each Term

$-3 \times 7 = -21$ and $-3 \times 5g = -15g$.
Combining these, we get $-21 - 15g$.

Step 1: Remove Parentheses (Associative Property of Addition)

The associative property of addition allows us to group terms differently. Since addition is associative, $(x + 7) + 3y = x + 7 + 3y$ (or we can also write it as $x + 3y + 7$ by commutative property, but the simplest equivalent is just removing the parentheses as is).

Step 1: Use Commutative Property of Addition

The commutative property of addition states that $a + b = b + a$. So we can rearrange the terms: $\frac{2}{9} + x - \frac{1}{5}$ or $x + \frac{2}{9} - \frac{1}{5}$. Alternatively, we can combine the fractions first (but since the problem just asks for an equivalent expression, rearranging is sufficient).

Step 2 (Optional: Combine Fractions):

To combine $\frac{2}{9}$ and $-\frac{1}{5}$, find a common denominator (45).
$\frac{2}{9} = \frac{10}{45}$, $-\frac{1}{5} = -\frac{9}{45}$. So $\frac{10}{45} - \frac{9}{45} + x = \frac{1}{45} + x$.

Answer:

$-21 - 15g$

Problem 8: $(x + 7) + 3y$