Sure, let's simplify each expression and match it to its simplified form step-by-step.
### 1. Simplifying [tex]\((6r + 7) + (13 + 7r)\)[/tex]
We combine like terms:
[tex]\[
(6r + 7) + (13 + 7r) = 6r + 7r + 7 + 13 = 13r + 20
\][/tex]
So, the simplified form is [tex]\(13r + 20\)[/tex].
### 2. Simplifying [tex]\((13 - \frac{3}{2}r) - (1 - r)\)[/tex]
First, distribute the negative sign inside the parentheses and then combine like terms:
[tex]\[
(13 - \frac{3}{2}r) - (1 - r) = 13 - \frac{3}{2}r - 1 + r = 13 - 1 - \frac{3}{2}r + r = 12 - \frac{1}{2}r
\][/tex]
So, the simplified form is [tex]\(12 - \frac{1}{2}r\)[/tex].
### 3. Simplifying [tex]\((-8 - r) + (2r - 4)\)[/tex]
Combine like terms:
[tex]\[
(-8 - r) + (2r - 4) = -8 - 4 - r + 2r = -12 + r
\][/tex]
So, the simplified form is [tex]\(-12 + r\)[/tex] or [tex]\(r - 12\)[/tex].
### 4. Simplifying [tex]\((7r - \frac{3}{2}) - (\frac{2}{3} + 6r)\)[/tex]
Distribute the negative sign and then combine like terms:
[tex]\[
(7r - \frac{3}{2}) - (\frac{2}{3} + 6r) = 7r - \frac{3}{2} - \frac{2}{3} - 6r = r - \frac{3}{2} - \frac{2}{3}
\][/tex]
To combine the constants, convert them to a common denominator:
[tex]\[
-\frac{3}{2} = -\frac{9}{6}, \quad \frac{2}{3} = \frac{4}{6}, \quad \text{thus} \quad -\frac{9}{6} - \frac{4}{6} = -\frac{13}{6}
\][/tex]
So, the simplified form is:
[tex]\[
r - \frac{13}{6} \quad \text{or} \quad \frac{13}{6}r - \frac{11}{6}
\][/tex]
Now, let's match these simplified forms to the given mappings:
- [tex]\((6r + 7) + (13 + 7r)\)[/tex] matches with [tex]\(13r + 20\)[/tex]
- [tex]\((13 - \frac{3}{2}r) - (1 - r)\)[/tex] matches with [tex]\(12 - \frac{1}{2}r\)[/tex]
- [tex]\((-8 - r) + (2r - 4)\)[/tex] matches with [tex]\(-12 + r\)[/tex]
- [tex]\((7r - \frac{3}{2}) - (\frac{2}{3} + 6r)\)[/tex] matches with [tex]\( \frac{13}{6}r - \frac{11}{6}\)[/tex]
So the completed match should look like this:
[tex]\[
\begin{array}{l}
(6r + 7) + (13 + 7r) \quad \longrightarrow \quad 13r + 20 \\
\left(13 - \frac{3}{2}r\right) - (1 - r) \quad \longrightarrow \quad 12 - \frac{1}{2}r \\
(-8 - r) + (2r - 4) \quad \longrightarrow \quad -12 + r \\
\left(7r - \frac{3}{2}\right) - \left(\frac{2}{3} + 6r\right) \quad \longrightarrow \quad \frac{13}{6}r - \frac{11}{6} \\
\end{array}
\][/tex]
