Answer :
To solve the problem, we need to find two statements that, when combined, are equivalent to the original biconditional statement:
[tex]\[ 4r + 2 = 4 \text{ if and only if } 4s - 5 = 0. \][/tex]
A biconditional statement [tex]\( p \iff q \)[/tex] means that both [tex]\( p \implies q \)[/tex] and [tex]\( q \implies p \)[/tex] must be true. Thus, we need to identify the two conditional statements that represent this relationship.
1. If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s - 5 = 0 \)[/tex].
2. If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s = -5 \)[/tex].
3. If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 4 = 2 \)[/tex].
4. If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 2 = 4 \)[/tex].
Let's examine these options step-by-step.
### Step-by-step Analysis:
Analysis of each statement:
- Statement 1:
- If [tex]\( 4r + 2 = 4 \)[/tex], then simplify this equation:
[tex]\[ 4r + 2 = 4 \implies 4r = 2 \implies r = 0.5. \][/tex]
- According to the conditional statement [tex]\( 4r + 2 = 4 \implies 4s - 5 = 0 \)[/tex],
- Substituting [tex]\( r = 0.5 \)[/tex]:
[tex]\[ r = 0.5 \implies 4(0.5) + 2 = 2 + 2 = 4 \implies \text{True}.\][/tex]
- So, [tex]\( 4s - 5 = 0 \)[/tex]:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- This is a valid implication.
- Statement 2:
- If [tex]\( 4r + 2 = 4 \)[/tex], then simplify this equation:
[tex]\[ 4r + 2 = 4 \implies 4r = 2 \implies r = 0.5. \][/tex]
- According to the conditional statement [tex]\( 4r + 2 = 4 \implies 4s = -5 \)[/tex]:
- Substituting [tex]\( r = 0.5 \)[/tex]:
[tex]\[ r = 0.5 \implies 4(0.5) + 2 = 2 + 2 = 4 \implies \text{True}.\][/tex]
- So, [tex]\( 4s = -5 \)[/tex] is incorrect because it contradicts the requirement from the biconditional [tex]\(4s - 5 = 0 \implies 4s = 5 \implies s = 1.25 \)[/tex].
- Statement 3:
- If [tex]\( 4s - 5 = 0 \)[/tex], then simplify this equation:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- According to the conditional statement [tex]\( 4s - 5 = 0 \implies 4r + 4 = 2 \)[/tex]:
- Substituting [tex]\( s = 1.25 \)[/tex]:
[tex]\[ s = 1.25 \implies 4(1.25) - 5 = 5 - 5 = 0 \implies \text{True}.\][/tex]
- So, [tex]\( 4r + 4 = 2 \)[/tex] is incorrect as it is not structurally related to the original condition.
- Statement 4:
- If [tex]\( 4s - 5 = 0 \)[/tex], then simplify this equation:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- According to the conditional statement [tex]\( 4s - 5 = 0 \implies 4r + 2 = 4 \)[/tex]:
- Substituting [tex]\( s = 1.25 \)[/tex]:
[tex]\[ s = 1.25 \implies 4(1.25) - 5 = 5 - 5 = 0 \implies \text{True}.\][/tex]
- So, using [tex]\( s = 1.25 \)[/tex]:
[tex]\[ 4r + 2 = 4 \text{ holds true when } 4(0.5) + 2 = 4. \][/tex]
- Thus, this valid implication shows [tex]\( s = 1.25 \)[/tex] when [tex]\( r = 0.5 \)[/tex].
### Conclusion:
The two statements matching the biconditional [tex]\( 4r + 2 = 4 \text{ if and only if } 4s - 5 = 0 \)[/tex] by representing both directions are:
- Statement 1: If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s - 5 = 0 \)[/tex].
- Statement 4: If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 2 = 4 \)[/tex].
These satisfy the definitions for a biconditional.
[tex]\[ 4r + 2 = 4 \text{ if and only if } 4s - 5 = 0. \][/tex]
A biconditional statement [tex]\( p \iff q \)[/tex] means that both [tex]\( p \implies q \)[/tex] and [tex]\( q \implies p \)[/tex] must be true. Thus, we need to identify the two conditional statements that represent this relationship.
1. If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s - 5 = 0 \)[/tex].
2. If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s = -5 \)[/tex].
3. If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 4 = 2 \)[/tex].
4. If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 2 = 4 \)[/tex].
Let's examine these options step-by-step.
### Step-by-step Analysis:
Analysis of each statement:
- Statement 1:
- If [tex]\( 4r + 2 = 4 \)[/tex], then simplify this equation:
[tex]\[ 4r + 2 = 4 \implies 4r = 2 \implies r = 0.5. \][/tex]
- According to the conditional statement [tex]\( 4r + 2 = 4 \implies 4s - 5 = 0 \)[/tex],
- Substituting [tex]\( r = 0.5 \)[/tex]:
[tex]\[ r = 0.5 \implies 4(0.5) + 2 = 2 + 2 = 4 \implies \text{True}.\][/tex]
- So, [tex]\( 4s - 5 = 0 \)[/tex]:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- This is a valid implication.
- Statement 2:
- If [tex]\( 4r + 2 = 4 \)[/tex], then simplify this equation:
[tex]\[ 4r + 2 = 4 \implies 4r = 2 \implies r = 0.5. \][/tex]
- According to the conditional statement [tex]\( 4r + 2 = 4 \implies 4s = -5 \)[/tex]:
- Substituting [tex]\( r = 0.5 \)[/tex]:
[tex]\[ r = 0.5 \implies 4(0.5) + 2 = 2 + 2 = 4 \implies \text{True}.\][/tex]
- So, [tex]\( 4s = -5 \)[/tex] is incorrect because it contradicts the requirement from the biconditional [tex]\(4s - 5 = 0 \implies 4s = 5 \implies s = 1.25 \)[/tex].
- Statement 3:
- If [tex]\( 4s - 5 = 0 \)[/tex], then simplify this equation:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- According to the conditional statement [tex]\( 4s - 5 = 0 \implies 4r + 4 = 2 \)[/tex]:
- Substituting [tex]\( s = 1.25 \)[/tex]:
[tex]\[ s = 1.25 \implies 4(1.25) - 5 = 5 - 5 = 0 \implies \text{True}.\][/tex]
- So, [tex]\( 4r + 4 = 2 \)[/tex] is incorrect as it is not structurally related to the original condition.
- Statement 4:
- If [tex]\( 4s - 5 = 0 \)[/tex], then simplify this equation:
[tex]\[ 4s - 5 = 0 \implies 4s = 5 \implies s = 1.25. \][/tex]
- According to the conditional statement [tex]\( 4s - 5 = 0 \implies 4r + 2 = 4 \)[/tex]:
- Substituting [tex]\( s = 1.25 \)[/tex]:
[tex]\[ s = 1.25 \implies 4(1.25) - 5 = 5 - 5 = 0 \implies \text{True}.\][/tex]
- So, using [tex]\( s = 1.25 \)[/tex]:
[tex]\[ 4r + 2 = 4 \text{ holds true when } 4(0.5) + 2 = 4. \][/tex]
- Thus, this valid implication shows [tex]\( s = 1.25 \)[/tex] when [tex]\( r = 0.5 \)[/tex].
### Conclusion:
The two statements matching the biconditional [tex]\( 4r + 2 = 4 \text{ if and only if } 4s - 5 = 0 \)[/tex] by representing both directions are:
- Statement 1: If [tex]\( 4r + 2 = 4 \)[/tex], then [tex]\( 4s - 5 = 0 \)[/tex].
- Statement 4: If [tex]\( 4s - 5 = 0 \)[/tex], then [tex]\( 4r + 2 = 4 \)[/tex].
These satisfy the definitions for a biconditional.
