Consider the system of equations:
[tex]\[
\left\{\begin{array}{l}
x - 4y = 6 \\
3x + y = -8
\end{array}\right.
\][/tex]
We need to determine whether the ordered pair [tex]\((-3, 1)\)[/tex] fits each equation and the system as a whole.
### Step 1: Verify the first equation [tex]\(x - 4y = 6\)[/tex]
Substitute [tex]\(x = -3\)[/tex] and [tex]\(y = 1\)[/tex] into [tex]\(x - 4y = 6\)[/tex]:
[tex]\[
-3 - 4(1) = -3 - 4 = -7
\][/tex]
Since [tex]\( -7 \neq 6 \)[/tex], the ordered pair [tex]\((-3, 1)\)[/tex] does not satisfy the first equation. Therefore, the statement "The ordered pair [tex]\((-3, 1)\)[/tex] is a solution to the first equation because it makes the first equation true" is false.
### Step 2: Verify the second equation [tex]\(3x + y = -8\)[/tex]
Substitute [tex]\(x = -3\)[/tex] and [tex]\(y = 1\)[/tex] into [tex]\(3x + y = -8\)[/tex]:
[tex]\[
3(-3) + 1 = -9 + 1 = -8
\][/tex]
Since [tex]\(-8 = -8\)[/tex], the ordered pair [tex]\((-3, 1)\)[/tex] does satisfy the second equation. Therefore, the statement "The ordered pair [tex]\((-3, 1)\)[/tex] is a solution to the second equation because it makes the second equation true" is true.
### Step 3: Verify the system of equations
To determine if [tex]\((-3, 1)\)[/tex] is a solution to the system, it must satisfy both equations. From the verification steps:
- The ordered pair [tex]\((-3, 1)\)[/tex] does not satisfy the first equation.
- The ordered pair [tex]\((-3, 1)\)[/tex] satisfies the second equation.
Since it does not satisfy both equations, the ordered pair [tex]\((-3, 1)\)[/tex] is not a solution to the system. Therefore, the statement "The ordered pair [tex]\((-3, 1)\)[/tex] is not a solution to the system because it makes at least one of the equations false" is true, and the statement "The ordered pair [tex]\((-3, 1)\)[/tex] is a solution to the system because it makes both equations true" is false.
### Conclusion
The correct statements for the ordered pair [tex]\((-3, 1)\)[/tex] and the system of equations are:
- The ordered pair [tex]\((-3, 1)\)[/tex] is a solution to the second equation because it makes the second equation true.
- The ordered pair [tex]\((-3, 1)\)[/tex] is not a solution to the system because it makes at least one of the equations false.
