Get expert insights and reliable answers to your questions on IDNLearn.com. Discover detailed answers to your questions with our extensive database of expert knowledge.

Which ordered pair is the solution to the system of equations?

[tex]\[
\left\{
\begin{array}{l}
x + 2y = -8 \\
y = \frac{1}{2}x - 2
\end{array}
\right.
\][/tex]

A. [tex]\((-2, -1)\)[/tex]
B. [tex]\((-2, -3)\)[/tex]
C. [tex]\((-6, -5)\)[/tex]
D. [tex]\((-4, -4)\)[/tex]


Sagot :

To determine which ordered pair is the solution to the system of equations
[tex]\[ \left\{\begin{array}{l} x + 2y = -8 \\ y = \frac{1}{2}x - 2 \end{array}\right., \][/tex]
we need to verify which pair satisfies both equations.

Let's analyze each pair one by one.

### Pair [tex]\((-2, -1)\)[/tex]

1. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -1\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[ -2 + 2(-1) = -2 - 2 = -4 \quad (\text{not } -8) \][/tex]
Hence, this pair does not satisfy the first equation.

### Pair [tex]\((-2, -3)\)[/tex]

1. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -3\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[ -2 + 2(-3) = -2 - 6 = -8 \quad (\text{correct}) \][/tex]
2. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -3\)[/tex] into the second equation [tex]\(y = \frac{1}{2}x - 2\)[/tex]:
[tex]\[ -3 = \frac{1}{2}(-2) - 2 = -1 - 2 = -3 \quad (\text{correct}) \][/tex]
This pair satisfies both equations.

### Pair [tex]\((-6, -5)\)[/tex]

1. Substitute [tex]\(x = -6\)[/tex] and [tex]\(y = -5\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[ -6 + 2(-5) = -6 - 10 = -16 \quad (\text{not } -8) \][/tex]
Hence, this pair does not satisfy the first equation.

### Pair [tex]\((-4, -4)\)[/tex]

1. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -4\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[ -4 + 2(-4) = -4 - 8 = -12 \quad (\text{not } -8) \][/tex]
Hence, this pair does not satisfy the first equation.

Based on the above evaluations, the only ordered pair that satisfies both equations is:
[tex]\[ (-2, -3) \][/tex]

Thus, the solution to the system of equations is [tex]\(\boxed{(-2, -3)}\)[/tex].