Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Solve the following system of equations and inequalities:
[tex]\[
\begin{cases}
2x + y = 4 \\
x + 2y \leq 8
\end{cases}
\][/tex]

Sagot :

GinnyAnswer
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).
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.