Which statements are true about the ordered pair [tex]$(-4,0)$[/tex] and the system of equations?

[tex]
\left\{
\begin{array}{l}
2x + y = -8 \\
x - y = -4
\end{array}
\right.
[/tex]

Select each correct answer.

A. The ordered pair [tex]$(-4,0)$[/tex] is a solution to the first equation because it makes the first equation true.

B. The ordered pair [tex][tex]$(-4,0)$[/tex][/tex] is a solution to the second equation because it makes the second equation true.

C. The ordered pair [tex]$(-4,0)$[/tex] is not a solution to the system because it makes at least one of the equations false.

D. The ordered pair [tex]$(-4,0)$[/tex] is a solution to the system because it makes both equations true.



Answer :

Given the ordered pair [tex]\((-4, 0)\)[/tex] and the system of equations:
[tex]\[ \left\{\begin{array}{l} 2x + y = -8 \\ x - y = -4 \end{array}\right. \][/tex]

Let's evaluate each equation:

### First Equation: [tex]\(2x + y = -8\)[/tex]

Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 0\)[/tex] into the first equation:
[tex]\[ 2(-4) + 0 = -8 \][/tex]
[tex]\[ -8 = -8 \][/tex]
The ordered pair [tex]\((-4, 0)\)[/tex] satisfies the first equation because the left-hand side equals the right-hand side. So, the statement "The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the first equation because it makes the first equation true." is true.

### Second Equation: [tex]\(x - y = -4\)[/tex]

Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 0\)[/tex] into the second equation:
[tex]\[ -4 - 0 = -4 \][/tex]
[tex]\[ -4 = -4 \][/tex]
The ordered pair [tex]\((-4, 0)\)[/tex] satisfies the second equation because the left-hand side equals the right-hand side. So, the statement "The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the second equation because it makes the second equation true." is true.

### Solution to the System

Since [tex]\((-4, 0)\)[/tex] satisfies both equations, it is a solution to the system. Therefore, the statement "The ordered pair [tex]\((-4,0)\)[/tex] is not a solution to the system because it makes at least one of the equations false." is false.

Conversely, the statement "The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the system because it makes both equations true." is true.

### Conclusion

The correct statements about the ordered pair [tex]\((-4,0)\)[/tex] and the system of equations are:
1. The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the first equation because it makes the first equation true.
2. The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the second equation because it makes the second equation true.
3. The ordered pair [tex]\((-4,0)\)[/tex] is a solution to the system because it makes both equations true.