Answer :
To solve the given system of equations and inequality, we need to follow these steps:
### Step 1: Solve the equality for \( y \)
We are given the equality:
[tex]\[ 2x + y = 4 \][/tex]
To solve for \( y \), we isolate \( y \) on one side of the equation:
[tex]\[ y = 4 - 2x \][/tex]
### Step 2: Substitute \( y \) into the inequality
We now substitute \( y = 4 - 2x \) into the inequality to check the constraint:
[tex]\[ x + 2y \leq 8 \][/tex]
Substitute \( y \):
[tex]\[ x + 2(4 - 2x) \leq 8 \][/tex]
Simplify the inequality:
[tex]\[ x + 8 - 4x \leq 8 \][/tex]
[tex]\[ -3x + 8 \leq 8 \][/tex]
Subtract 8 from both sides:
[tex]\[ -3x \leq 0 \][/tex]
Divide by -3 (remember to reverse the inequality sign):
[tex]\[ x \geq 0 \][/tex]
### Step 3: Determine the feasible region for \( x \)
The feasible region for \( x \) is:
[tex]\[ 0 \leq x < \infty \][/tex]
### Final Answer:
The solution to the system consists of the expression for \( y \) and the feasible region for \( x \):
[tex]\[ y = 4 - 2x \][/tex]
[tex]\[ 0 \leq x < \infty \][/tex]
This means that [tex]\( y = 4 - 2x \)[/tex] is valid for all [tex]\( x \)[/tex] in the range from 0 to infinity (not including infinity).
### Step 1: Solve the equality for \( y \)
We are given the equality:
[tex]\[ 2x + y = 4 \][/tex]
To solve for \( y \), we isolate \( y \) on one side of the equation:
[tex]\[ y = 4 - 2x \][/tex]
### Step 2: Substitute \( y \) into the inequality
We now substitute \( y = 4 - 2x \) into the inequality to check the constraint:
[tex]\[ x + 2y \leq 8 \][/tex]
Substitute \( y \):
[tex]\[ x + 2(4 - 2x) \leq 8 \][/tex]
Simplify the inequality:
[tex]\[ x + 8 - 4x \leq 8 \][/tex]
[tex]\[ -3x + 8 \leq 8 \][/tex]
Subtract 8 from both sides:
[tex]\[ -3x \leq 0 \][/tex]
Divide by -3 (remember to reverse the inequality sign):
[tex]\[ x \geq 0 \][/tex]
### Step 3: Determine the feasible region for \( x \)
The feasible region for \( x \) is:
[tex]\[ 0 \leq x < \infty \][/tex]
### Final Answer:
The solution to the system consists of the expression for \( y \) and the feasible region for \( x \):
[tex]\[ y = 4 - 2x \][/tex]
[tex]\[ 0 \leq x < \infty \][/tex]
This means that [tex]\( y = 4 - 2x \)[/tex] is valid for all [tex]\( x \)[/tex] in the range from 0 to infinity (not including infinity).
