Answer :
To match the expressions that are equivalent, we need to simplify each one step-by-step and compare them.
1. Expression 1: [tex]\((-14+\frac{3}{2} b)-(1+\frac{8}{2} b)\)[/tex]
[tex]\[ = -14 + \frac{3}{2} b - 1 - 4b = -15 + \frac{3}{2} b - 4b = -15 + \frac{3}{2} b - \frac{8}{2} b = -15 - \frac{5}{2} b \][/tex]
2. Expression 2: [tex]\(4 b + \frac{13}{2}\)[/tex]
This expression is already in its simplest form.
3. Expression 3: [tex]\((5 + 2b) + \left(2b + \frac{3}{2}\right)\)[/tex]
[tex]\[ = 5 + 2b + 2b + \frac{3}{2} = 5 + 4b + \frac{3}{2} = 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
4. Expression 4: [tex]\(8 b - 15\)[/tex]
This expression is already in its simplest form.
5. Expression 5: \left(\frac{7}{2} b-3\right)-(8+6 b\)
[tex]\[ = \frac{7}{2} b - 3 - 8 - 6b = \frac{7}{2} b - 6b - 11 = \frac{7}{2} b - \frac{12}{2} b - 11 = -\frac{5}{2} b - 11 \][/tex]
6. Expression 6: [tex]\((-10+b)+(7 b-5)\)[/tex]
[tex]\[ = -10 + b + 7b - 5 = -10 - 5 + 8b = -15 + 8b \][/tex]
7. Expression 7: [tex]\(-15 - \frac{5}{2} b\)[/tex]
This expression is already in its simplest form.
Now we can match the pairs:
- Expression 1 [tex]\((-14+\frac{3}{2} b)-(1+\frac{8}{2} b)\)[/tex] is equivalent to Expression 7 [tex]\(-15 - \frac{5}{2} b\)[/tex]
- Expression 2 [tex]\(4 b + \frac{13}{2}\)[/tex] is equivalent to Expression 3 [tex]\((5 + 2 b) + \left(2 b + \frac{3}{2}\right)\)[/tex]
- Expression 4 [tex]\(8 b - 15\)[/tex] is equivalent to Expression 6 [tex]\((-10+b)+(7 b-5)\)[/tex]
- Expression 5 [tex]\(\left(\frac{7}{2} b-3\right)-(8+6 b)\)[/tex] is already in its simple form, so no pair is mentioned (solitary).
Thus, the correct pairs are:
1. [tex]\( (-14+\frac{3}{2} b)-(1+\frac{8}{2} b) \)[/tex] and [tex]\( -15-\frac{5}{2} b \)[/tex]
2. [tex]\( 4 b + \frac{13}{2} \)[/tex] and [tex]\( (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \)[/tex]
3. [tex]\( 8 b - 15 \)[/tex] and [tex]\( (-10+b)+(7 b-5) \)[/tex]
4. [tex]\( \left(\frac{7}{2} b-3\right)-(8+6 b) \)[/tex] (unpaired)
These are the equivalent pairs of expressions.
1. Expression 1: [tex]\((-14+\frac{3}{2} b)-(1+\frac{8}{2} b)\)[/tex]
[tex]\[ = -14 + \frac{3}{2} b - 1 - 4b = -15 + \frac{3}{2} b - 4b = -15 + \frac{3}{2} b - \frac{8}{2} b = -15 - \frac{5}{2} b \][/tex]
2. Expression 2: [tex]\(4 b + \frac{13}{2}\)[/tex]
This expression is already in its simplest form.
3. Expression 3: [tex]\((5 + 2b) + \left(2b + \frac{3}{2}\right)\)[/tex]
[tex]\[ = 5 + 2b + 2b + \frac{3}{2} = 5 + 4b + \frac{3}{2} = 4b + 5 + \frac{3}{2} = 4b + \frac{10}{2} + \frac{3}{2} = 4b + \frac{13}{2} \][/tex]
4. Expression 4: [tex]\(8 b - 15\)[/tex]
This expression is already in its simplest form.
5. Expression 5: \left(\frac{7}{2} b-3\right)-(8+6 b\)
[tex]\[ = \frac{7}{2} b - 3 - 8 - 6b = \frac{7}{2} b - 6b - 11 = \frac{7}{2} b - \frac{12}{2} b - 11 = -\frac{5}{2} b - 11 \][/tex]
6. Expression 6: [tex]\((-10+b)+(7 b-5)\)[/tex]
[tex]\[ = -10 + b + 7b - 5 = -10 - 5 + 8b = -15 + 8b \][/tex]
7. Expression 7: [tex]\(-15 - \frac{5}{2} b\)[/tex]
This expression is already in its simplest form.
Now we can match the pairs:
- Expression 1 [tex]\((-14+\frac{3}{2} b)-(1+\frac{8}{2} b)\)[/tex] is equivalent to Expression 7 [tex]\(-15 - \frac{5}{2} b\)[/tex]
- Expression 2 [tex]\(4 b + \frac{13}{2}\)[/tex] is equivalent to Expression 3 [tex]\((5 + 2 b) + \left(2 b + \frac{3}{2}\right)\)[/tex]
- Expression 4 [tex]\(8 b - 15\)[/tex] is equivalent to Expression 6 [tex]\((-10+b)+(7 b-5)\)[/tex]
- Expression 5 [tex]\(\left(\frac{7}{2} b-3\right)-(8+6 b)\)[/tex] is already in its simple form, so no pair is mentioned (solitary).
Thus, the correct pairs are:
1. [tex]\( (-14+\frac{3}{2} b)-(1+\frac{8}{2} b) \)[/tex] and [tex]\( -15-\frac{5}{2} b \)[/tex]
2. [tex]\( 4 b + \frac{13}{2} \)[/tex] and [tex]\( (5 + 2 b) + \left(2 b + \frac{3}{2}\right) \)[/tex]
3. [tex]\( 8 b - 15 \)[/tex] and [tex]\( (-10+b)+(7 b-5) \)[/tex]
4. [tex]\( \left(\frac{7}{2} b-3\right)-(8+6 b) \)[/tex] (unpaired)
These are the equivalent pairs of expressions.
