Let's solve the system of equations using the method of substitution. Here are the equations:
[tex]\[
\begin{cases}
-2 x + y = -5 \\
-4 y = 22 - 8 x
\end{cases}
\][/tex]
### Step 1: Solve one of the equations for one variable
We start by solving the first equation for [tex]\( y \)[/tex]:
[tex]\[
-2 x + y = -5 \implies y = 2x - 5
\][/tex]
### Step 2: Substitute the expression for [tex]\( y \)[/tex] into the second equation
Next, we substitute [tex]\( y = 2x - 5 \)[/tex] into the second equation:
[tex]\[
-4 (2x - 5) = 22 - 8x
\][/tex]
### Step 3: Simplify the substituted equation and solve for [tex]\( x \)[/tex]
First, expand and simplify the equation:
[tex]\[
-4 (2x - 5) = 22 - 8x \\
-8x + 20 = 22 - 8x
\][/tex]
Notice that the terms [tex]\(-8x\)[/tex] on both sides cancel each other out:
[tex]\[
20 = 22
\][/tex]
This is a contradiction because 20 does not equal 22. Therefore, the system of equations has no solution.
### Conclusion:
Since we have derived a contradiction, it indicates that the system of equations is inconsistent and there are no solutions.
[tex]\[
\text{No solutions}
\][/tex]
