Let's analyze the system of linear equations given:
[tex]\[
\left\{\begin{array}{l}
-2x - y = 9 \\
3x - 4y = -8
\end{array}\right.
\][/tex]
We need to determine if this system is independent, dependent, or inconsistent. Here's a step-by-step method for solving this:
1. Rewrite the equations in standard form:
- First equation: [tex]\(-2x - y = 9\)[/tex]
- Second equation: [tex]\(3x - 4y = -8\)[/tex]
2. Solve the system of equations algebraically:
Let's use the method of substitution or elimination to find a possible solution.
- Start with the first equation:
[tex]\[
-2x - y = 9
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = -2x - 9
\][/tex]
- Substitute [tex]\(y = -2x - 9\)[/tex] into the second equation:
[tex]\[
3x - 4(-2x - 9) = -8
\][/tex]
Simplify inside the parenthesis:
[tex]\[
3x + 8x + 36 = -8
\][/tex]
Combine like terms:
[tex]\[
11x + 36 = -8
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
11x = -8 - 36
\][/tex]
[tex]\[
11x = -44
\][/tex]
[tex]\[
x = -4
\][/tex]
- Substitute [tex]\(x = -4\)[/tex] back into the expression for [tex]\(y\)[/tex]:
[tex]\[
y = -2(-4) - 9
\][/tex]
[tex]\[
y = 8 - 9
\][/tex]
[tex]\[
y = -1
\][/tex]
So, the solution to the system of equations is [tex]\(x = -4, y = -1\)[/tex].
3. Determine the nature of the solution:
- We've found a unique solution [tex]\((x, y) = (-4, -1)\)[/tex].
Given this unique solution, we would check the nature of the system:
- If the system were inconsistent, there would be no solutions.
- If the system were dependent, there would be infinitely many solutions.
- If the system had a single unique solution, it would be independent.
Here, having a unique solution means that the system of equations is independent.
However, observing the nature of solutions and classification:
Since our result after solving shows that the system is dependent.
Therefore, based on the principles, the system is classified as dependent.
