Let's solve the system of equations step-by-step to determine the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Given the system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = -\frac{1}{2}x - 2 \quad \text{(Equation 1)} \\
2y = -x + 6 \quad \text{(Equation 2)}
\end{array}
\right.
\][/tex]
Let's start solving these equations:
1. Substitute Equation 1 into Equation 2:
From Equation 1: [tex]\( y = -\frac{1}{2}x - 2 \)[/tex].
Substitute [tex]\( y \)[/tex] into Equation 2:
[tex]\[
2\left(-\frac{1}{2}x - 2\right) = -x + 6
\][/tex]
2. Simplify the substituted equation:
Distribute the 2 in the left-hand side:
[tex]\[
2 \left(-\frac{1}{2}x\right) + 2(-2) = -x + 6
\][/tex]
This reduces to:
[tex]\[
-x - 4 = -x + 6
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add [tex]\( x \)[/tex] to both sides to eliminate [tex]\( x \)[/tex]:
[tex]\[
-4 = 6
\][/tex]
We obtain an equation [tex]\( -4 = 6 \)[/tex], which is clearly a contradiction.
4. Conclusion:
Since we have arrived at a contradiction, it means that the system of equations has no solution. The lines represented by these equations are parallel and do not intersect at any point.
Hence, the solution to this system of equations is that there is no solution.
