Answer :
To match the pairs of equivalent expressions, we need to simplify each of the given mathematical expressions. Let's simplify them one by one.
1. Simplify the expression:
[tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \][/tex]
First, simplify inside the parentheses:
[tex]\[ -14 + \frac{3}{2} b \][/tex]
[tex]\[ 1 + \frac{8}{2} b = 1 + 4b \][/tex]
Now, subtract the two simplified expressions:
[tex]\[ -14 + \frac{3}{2} b - 1 - 4b \][/tex]
Combine the terms:
[tex]\[ -15 + \frac{3}{2} b - 4b \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ -15 + \frac{3}{2} b - \frac{8}{2} b = -15 - \frac{5}{2} b \][/tex]
2. Simplify the expression:
[tex]\[ (5 + 2b) + \left(2b + \frac{3}{2}\right) \][/tex]
First, simplify inside the parentheses:
[tex]\[ 5 + 2b \][/tex]
[tex]\[ 2b + \frac{3}{2} \][/tex]
Now, add the two simplified expressions:
[tex]\[ 5 + 2b + 2b + \frac{3}{2} \][/tex]
Combine the terms:
[tex]\[ 5 + 4b + \frac{3}{2} \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ 5 + \frac{10}{2} + 4b + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
3. Simplify the expression:
[tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6b) \][/tex]
First, simplify inside the parentheses:
[tex]\[ \frac{7}{2} b - 3 \][/tex]
[tex]\[ 8 + 6b \][/tex]
Now, subtract the two simplified expressions:
[tex]\[ \frac{7}{2} b - 3 - 8 - 6b \][/tex]
Combine the terms:
[tex]\[ \frac{7}{2} b - 6b - 3 - 8 \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ \frac{7}{2} b - \frac{12}{2} b - 11 = -\frac{5}{2} b - 11 \][/tex]
4. Simplify the expression:
[tex]\[ (-10 + b) + (7b - 5) \][/tex]
First, simplify inside the parentheses:
[tex]\[ -10 + b \][/tex]
[tex]\[ 7b - 5 \][/tex]
Now, add the two simplified expressions:
[tex]\[ -10 + b + 7b - 5 \][/tex]
Combine the terms:
[tex]\[ -10 - 5 + b + 7b = 8b - 15 \][/tex]
Now, match these simplified expressions with the given expressions:
1. [tex]\((-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \Rightarrow -15 - \frac{5}{2} b\)[/tex]
2. [tex]\( (5 + 2b) + (2b + \frac{3}{2}) \Rightarrow 4 b + \frac{13}{2}\)[/tex]
3. [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b) \Rightarrow -\frac{5}{2} b - 11\)[/tex]
4. [tex]\((-10 + b) + (7b - 5) \Rightarrow 8 b - 15 \)[/tex]
Here are the matched pairs:
1. [tex]\((-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b)\)[/tex] with [tex]\(-15 - \frac{5}{2} b\)[/tex]
2. [tex]\( (5 + 2b) + (2b + \frac{3}{2})\)[/tex] with [tex]\(4 b + \frac{13}{2}\)[/tex]
3. [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b)\)[/tex] with [tex]\(-\frac{5}{2} b - 11\)[/tex]
4. [tex]\((-10 + b) + (7b - 5)\)[/tex] with [tex]\(8 b - 15\)[/tex]
1. Simplify the expression:
[tex]\[ \left(-14 + \frac{3}{2} b\right) - \left(1 + \frac{8}{2} b\right) \][/tex]
First, simplify inside the parentheses:
[tex]\[ -14 + \frac{3}{2} b \][/tex]
[tex]\[ 1 + \frac{8}{2} b = 1 + 4b \][/tex]
Now, subtract the two simplified expressions:
[tex]\[ -14 + \frac{3}{2} b - 1 - 4b \][/tex]
Combine the terms:
[tex]\[ -15 + \frac{3}{2} b - 4b \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ -15 + \frac{3}{2} b - \frac{8}{2} b = -15 - \frac{5}{2} b \][/tex]
2. Simplify the expression:
[tex]\[ (5 + 2b) + \left(2b + \frac{3}{2}\right) \][/tex]
First, simplify inside the parentheses:
[tex]\[ 5 + 2b \][/tex]
[tex]\[ 2b + \frac{3}{2} \][/tex]
Now, add the two simplified expressions:
[tex]\[ 5 + 2b + 2b + \frac{3}{2} \][/tex]
Combine the terms:
[tex]\[ 5 + 4b + \frac{3}{2} \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ 5 + \frac{10}{2} + 4b + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
3. Simplify the expression:
[tex]\[ \left(\frac{7}{2} b - 3\right) - (8 + 6b) \][/tex]
First, simplify inside the parentheses:
[tex]\[ \frac{7}{2} b - 3 \][/tex]
[tex]\[ 8 + 6b \][/tex]
Now, subtract the two simplified expressions:
[tex]\[ \frac{7}{2} b - 3 - 8 - 6b \][/tex]
Combine the terms:
[tex]\[ \frac{7}{2} b - 6b - 3 - 8 \][/tex]
Convert all terms to a common denominator if necessary:
[tex]\[ \frac{7}{2} b - \frac{12}{2} b - 11 = -\frac{5}{2} b - 11 \][/tex]
4. Simplify the expression:
[tex]\[ (-10 + b) + (7b - 5) \][/tex]
First, simplify inside the parentheses:
[tex]\[ -10 + b \][/tex]
[tex]\[ 7b - 5 \][/tex]
Now, add the two simplified expressions:
[tex]\[ -10 + b + 7b - 5 \][/tex]
Combine the terms:
[tex]\[ -10 - 5 + b + 7b = 8b - 15 \][/tex]
Now, match these simplified expressions with the given expressions:
1. [tex]\((-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b) \Rightarrow -15 - \frac{5}{2} b\)[/tex]
2. [tex]\( (5 + 2b) + (2b + \frac{3}{2}) \Rightarrow 4 b + \frac{13}{2}\)[/tex]
3. [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b) \Rightarrow -\frac{5}{2} b - 11\)[/tex]
4. [tex]\((-10 + b) + (7b - 5) \Rightarrow 8 b - 15 \)[/tex]
Here are the matched pairs:
1. [tex]\((-14 + \frac{3}{2} b) - (1 + \frac{8}{2} b)\)[/tex] with [tex]\(-15 - \frac{5}{2} b\)[/tex]
2. [tex]\( (5 + 2b) + (2b + \frac{3}{2})\)[/tex] with [tex]\(4 b + \frac{13}{2}\)[/tex]
3. [tex]\(\left(\frac{7}{2} b - 3\right) - (8 + 6b)\)[/tex] with [tex]\(-\frac{5}{2} b - 11\)[/tex]
4. [tex]\((-10 + b) + (7b - 5)\)[/tex] with [tex]\(8 b - 15\)[/tex]
