Solved on Feb 09, 2024

Find the missing operation in the equation $5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s$ to make it a true statement. Enter "+" for addition or "-" for subtraction.

STEP 1

Assumptions
1. We are given an equation with a blank indicating a missing operation.
2. We need to determine whether the operation is addition or subtraction to make the equation true.
3. The given equation is: $5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s]$

STEP 2

First, let's distribute the $5$ on the left side of the equation to each term inside the parentheses.
$5 \times \frac{1}{2} q + 5 \times 3 r - 5 \times 4 s - 9 r$

STEP 3

Now, perform the multiplication for each term.
$\frac{5}{2} q + 15 r - 20 s - 9 r$

STEP 4

Combine like terms on the left side of the equation by subtracting $9 r$ from $15 r$ .
$\frac{5}{2} q + (15 r - 9 r) - 20 s$

STEP 5

Simplify the like terms.
$\frac{5}{2} q + 6 r - 20 s$

STEP 6

Now we have the left side of the equation simplified. Let's write down the simplified left side and the original right side of the equation.
$\frac{5}{2} q + 6 r - 20 s = \frac{5}{2} q[\text { [blank] } 15 r - 20 s]$

STEP 7

To determine the operation in the blank, we need to compare the terms on both sides of the equation. We notice that the $\frac{5}{2} q$ and $-20 s$ terms are the same on both sides.

STEP 8

Now, compare the $r$ terms on both sides of the equation. On the left side, we have $6 r$ , and on the right side, we have a blank followed by $15 r$ .

STEP 9

To make the equation true, the operation in the blank should result in $6 r$ when combined with $15 r$ .

STEP 10

Determine the operation needed to transform $15 r$ into $6 r$ .
$15 r \text{ [blank] } = 6 r$

STEP 11

Subtract $15 r$ from both sides to isolate the blank operation.
$\text{ [blank] } = 6 r - 15 r$

STEP 12

Calculate the result of the operation.
$\text{ [blank] } = -9 r$

STEP 13

Since we have a negative result, the operation that transforms $15 r$ into $6 r$ is subtraction.
$\text{ [blank] } = -$
The operation that fills in the blank to make the equation a true statement is subtraction ("-").

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Find the missing operation in the equation $5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s$ to make it a true statement. Enter "+" for addition or "-" for subtraction.

STEP 1

Assumptions
1. We are given an equation with a blank indicating a missing operation.
2. We need to determine whether the operation is addition or subtraction to make the equation true.
3. The given equation is: $5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s]$

STEP 2

First, let's distribute the $5$ on the left side of the equation to each term inside the parentheses.
$5 \times \frac{1}{2} q + 5 \times 3 r - 5 \times 4 s - 9 r$

STEP 3

Now, perform the multiplication for each term.
$\frac{5}{2} q + 15 r - 20 s - 9 r$

STEP 4

Combine like terms on the left side of the equation by subtracting $9 r$ from $15 r$ .
$\frac{5}{2} q + (15 r - 9 r) - 20 s$

STEP 5

Simplify the like terms.
$\frac{5}{2} q + 6 r - 20 s$

STEP 6

Now we have the left side of the equation simplified. Let's write down the simplified left side and the original right side of the equation.
$\frac{5}{2} q + 6 r - 20 s = \frac{5}{2} q[\text { [blank] } 15 r - 20 s]$

STEP 7

To determine the operation in the blank, we need to compare the terms on both sides of the equation. We notice that the $\frac{5}{2} q$ and $-20 s$ terms are the same on both sides.

STEP 8

Now, compare the $r$ terms on both sides of the equation. On the left side, we have $6 r$ , and on the right side, we have a blank followed by $15 r$ .

STEP 9

To make the equation true, the operation in the blank should result in $6 r$ when combined with $15 r$ .

STEP 10

Determine the operation needed to transform $15 r$ into $6 r$ .
$15 r \text{ [blank] } = 6 r$

STEP 11

Subtract $15 r$ from both sides to isolate the blank operation.
$\text{ [blank] } = 6 r - 15 r$

STEP 12

Calculate the result of the operation.
$\text{ [blank] } = -9 r$

STEP 13

Since we have a negative result, the operation that transforms $15 r$ into $6 r$ is subtraction.
$\text{ [blank] } = -$
The operation that fills in the blank to make the equation a true statement is subtraction ("-").

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Find the missing operation in the equation 5(12q+3r−4s)−9r=52q[ [blank] 15r−20s5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s5(21​q+3r−4s)−9r=25​q[ [blank] 15r−20s to make it a true statement. Enter "+" for addition or "-" for subtraction.

STEP 1

STEP 2

First, let's distribute the 555 on the left side of the equation to each term inside the parentheses.5×12q+5×3r−5×4s−9r 5 \times \frac{1}{2} q + 5 \times 3 r - 5 \times 4 s - 9 r 5×21​q+5×3r−5×4s−9r

STEP 3

Now, perform the multiplication for each term.52q+15r−20s−9r \frac{5}{2} q + 15 r - 20 s - 9 r 25​q+15r−20s−9r

STEP 4

Combine like terms on the left side of the equation by subtracting 9r9 r9r from 15r15 r15r.52q+(15r−9r)−20s \frac{5}{2} q + (15 r - 9 r) - 20 s 25​q+(15r−9r)−20s

STEP 5

Simplify the like terms.52q+6r−20s \frac{5}{2} q + 6 r - 20 s 25​q+6r−20s

STEP 6

Now we have the left side of the equation simplified. Let's write down the simplified left side and the original right side of the equation.52q+6r−20s=52q[ [blank] 15r−20s] \frac{5}{2} q + 6 r - 20 s = \frac{5}{2} q[\text { [blank] } 15 r - 20 s] 25​q+6r−20s=25​q[ [blank] 15r−20s]

STEP 7

To determine the operation in the blank, we need to compare the terms on both sides of the equation. We notice that the 52q\frac{5}{2} q25​q and −20s-20 s−20s terms are the same on both sides.

STEP 8

Now, compare the rrr terms on both sides of the equation. On the left side, we have 6r6 r6r, and on the right side, we have a blank followed by 15r15 r15r.

STEP 9

To make the equation true, the operation in the blank should result in 6r6 r6r when combined with 15r15 r15r.

STEP 10

Determine the operation needed to transform 15r15 r15r into 6r6 r6r.15r [blank] =6r 15 r \text{ [blank] } = 6 r 15r [blank] =6r

STEP 11

Subtract 15r15 r15r from both sides to isolate the blank operation. [blank] =6r−15r \text{ [blank] } = 6 r - 15 r [blank] =6r−15r

STEP 12

Calculate the result of the operation. [blank] =−9r \text{ [blank] } = -9 r [blank] =−9r

STEP 13

Since we have a negative result, the operation that transforms 15r15 r15r into 6r6 r6r is subtraction. [blank] =− \text{ [blank] } = - [blank] =−The operation that fills in the blank to make the equation a true statement is subtraction ("-").

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Find the missing operation in the equation $5\left(\frac{1}{2} q+3 r-4 s\right)-9 r=\frac{5}{2} q[\text { [blank] } 15 r-20 s$ to make it a true statement. Enter "+" for addition or "-" for subtraction.

First, let's distribute the $5$ on the left side of the equation to each term inside the parentheses.
$5 \times \frac{1}{2} q + 5 \times 3 r - 5 \times 4 s - 9 r$

Now, perform the multiplication for each term.
$\frac{5}{2} q + 15 r - 20 s - 9 r$

Combine like terms on the left side of the equation by subtracting $9 r$ from $15 r$ .
$\frac{5}{2} q + (15 r - 9 r) - 20 s$

Simplify the like terms.
$\frac{5}{2} q + 6 r - 20 s$

Now we have the left side of the equation simplified. Let's write down the simplified left side and the original right side of the equation.
$\frac{5}{2} q + 6 r - 20 s = \frac{5}{2} q[\text { [blank] } 15 r - 20 s]$

To determine the operation in the blank, we need to compare the terms on both sides of the equation. We notice that the $\frac{5}{2} q$ and $-20 s$ terms are the same on both sides.

Now, compare the $r$ terms on both sides of the equation. On the left side, we have $6 r$ , and on the right side, we have a blank followed by $15 r$ .

To make the equation true, the operation in the blank should result in $6 r$ when combined with $15 r$ .

Determine the operation needed to transform $15 r$ into $6 r$ .
$15 r \text{ [blank] } = 6 r$

Subtract $15 r$ from both sides to isolate the blank operation.
$\text{ [blank] } = 6 r - 15 r$

Calculate the result of the operation.
$\text{ [blank] } = -9 r$

Since we have a negative result, the operation that transforms $15 r$ into $6 r$ is subtraction.
$\text{ [blank] } = -$
The operation that fills in the blank to make the equation a true statement is subtraction ("-").