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Sagot :
To analyze the given ordered pair [tex]\((-1, -4)\)[/tex] and determine which statements are true about how it fits into the given system of equations:
[tex]\[ \begin{aligned} &\text{1.} \quad x - y = 3 \\ &\text{2.} \quad 7x - y = -3 \end{aligned} \][/tex]
we will substitute [tex]\((-1, -4)\)[/tex] into each equation, and assess the truthfulness of each statement.
### Checking the Equations
Substitute [tex]\((-1, -4)\)[/tex] into the first equation:
[tex]\[ x - y = 3 \][/tex]
Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ -1 - (-4) = -1 + 4 = 3 \][/tex]
The left-hand side equals the right-hand side, so the ordered pair [tex]\((-1, -4)\)[/tex] satisfies the first equation. Therefore, statement 3 ("When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true.") is correct.
Substitute [tex]\((-1, -4)\)[/tex] into the second equation:
[tex]\[ 7x - y = -3 \][/tex]
Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ 7(-1) - (-4) = -7 + 4 = -3 \][/tex]
The left-hand side equals the right-hand side, so the ordered pair [tex]\((-1, -4)\)[/tex] satisfies the second equation. Therefore, statement 1 ("When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true.") is correct.
### Verifying Statements
Given that [tex]\((-1, -4)\)[/tex] satisfies both equations:
- Statement 1: When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true. (True)
- Statement 2: The ordered pair [tex]\((-1, -4)\)[/tex] is not a solution to the system of linear equations. (False; it satisfies both equations, so it is a solution)
- Statement 3: When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true. (True)
- Statement 4: When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is false. (False; substituting [tex]\((-1, -4)\)[/tex] into the first equation, it holds true)
- Statement 5: When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is false. (False; substituting [tex]\((-1, -4)\)[/tex] into the second equation, it holds true)
- Statement 6: The ordered pair [tex]\((-1, -4)\)[/tex] is a solution to the system of linear equations. (True)
So, the correct answers are:
1. When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true.
3. When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true.
6. The ordered pair [tex]\((-1, -4)\)[/tex] is a solution to the system of linear equations.
[tex]\[ \begin{aligned} &\text{1.} \quad x - y = 3 \\ &\text{2.} \quad 7x - y = -3 \end{aligned} \][/tex]
we will substitute [tex]\((-1, -4)\)[/tex] into each equation, and assess the truthfulness of each statement.
### Checking the Equations
Substitute [tex]\((-1, -4)\)[/tex] into the first equation:
[tex]\[ x - y = 3 \][/tex]
Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ -1 - (-4) = -1 + 4 = 3 \][/tex]
The left-hand side equals the right-hand side, so the ordered pair [tex]\((-1, -4)\)[/tex] satisfies the first equation. Therefore, statement 3 ("When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true.") is correct.
Substitute [tex]\((-1, -4)\)[/tex] into the second equation:
[tex]\[ 7x - y = -3 \][/tex]
Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ 7(-1) - (-4) = -7 + 4 = -3 \][/tex]
The left-hand side equals the right-hand side, so the ordered pair [tex]\((-1, -4)\)[/tex] satisfies the second equation. Therefore, statement 1 ("When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true.") is correct.
### Verifying Statements
Given that [tex]\((-1, -4)\)[/tex] satisfies both equations:
- Statement 1: When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true. (True)
- Statement 2: The ordered pair [tex]\((-1, -4)\)[/tex] is not a solution to the system of linear equations. (False; it satisfies both equations, so it is a solution)
- Statement 3: When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true. (True)
- Statement 4: When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is false. (False; substituting [tex]\((-1, -4)\)[/tex] into the first equation, it holds true)
- Statement 5: When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is false. (False; substituting [tex]\((-1, -4)\)[/tex] into the second equation, it holds true)
- Statement 6: The ordered pair [tex]\((-1, -4)\)[/tex] is a solution to the system of linear equations. (True)
So, the correct answers are:
1. When [tex]\((-1, -4)\)[/tex] is substituted into the second equation, the equation is true.
3. When [tex]\((-1, -4)\)[/tex] is substituted into the first equation, the equation is true.
6. The ordered pair [tex]\((-1, -4)\)[/tex] is a solution to the system of linear equations.
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