Connect with a global community of knowledgeable individuals on IDNLearn.com. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.
Sagot :
To solve the system of inequalities graphically and determine whether the solution region is bounded or unbounded, follow these steps:
### Step 1: Understanding the Inequalities
We have the following system of inequalities:
1. [tex]\( x + 2 y \leq 4 \)[/tex]
2. [tex]\( x \geq 0 \)[/tex]
3. [tex]\( y \geq 0 \)[/tex]
### Step 2: Plotting the Boundary Lines
First, consider the equality for each inequality:
1. [tex]\( x + 2 y = 4 \)[/tex]
2. [tex]\( x = 0 \)[/tex]
3. [tex]\( y = 0 \)[/tex]
- [tex]\( x + 2 y = 4 \)[/tex]:
This is a straight line. We can find the intercepts:
- For [tex]\( x \)[/tex]-intercept ([tex]\(y = 0\)[/tex]):
[tex]\( x + 2(0) = 4 \Rightarrow x = 4 \)[/tex]
- For [tex]\( y \)[/tex]-intercept ([tex]\(x = 0\)[/tex]):
[tex]\( 0 + 2 y = 4 \Rightarrow 2 y = 4 \Rightarrow y = 2 \)[/tex]
So, the line passes through points [tex]\( (4, 0) \)[/tex] and [tex]\( (0, 2) \)[/tex].
- [tex]\( x = 0 \)[/tex]:
This is the [tex]\( y \)[/tex]-axis.
- [tex]\( y = 0 \)[/tex]:
This is the [tex]\( x \)[/tex]-axis.
### Step 3: Shading the Regions
- For [tex]\( x + 2 y \leq 4 \)[/tex]: We shade the region below the line [tex]\( x + 2 y = 4 \)[/tex].
- For [tex]\( x \geq 0 \)[/tex]: We shade the region to the right of the [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]).
- For [tex]\( y \geq 0 \)[/tex]: We shade the region above the [tex]\( x \)[/tex]-axis ([tex]\( y = 0 \)[/tex]).
### Step 4: Determining the Solution Region
The solution region is the intersection of all these shaded regions. The feasible region is bounded by the lines [tex]\( x + 2 y = 4 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( y = 0 \)[/tex].
### Step 5: Identifying the Corner Points
The corner points of the feasible solution region (where lines intersect) can be calculated:
1. Intersection of [tex]\( x + 2 y = 4 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- Set [tex]\( x = 0 \)[/tex] in [tex]\( x + 2 y = 4 \)[/tex]:
[tex]\( 0 + 2 y = 4 \Rightarrow y = 2 \)[/tex]
- Coordinate: [tex]\( (0, 2) \)[/tex]
2. Intersection of [tex]\( x + 2 y = 4 \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex] in [tex]\( x + 2 y = 4 \)[/tex]:
[tex]\( x + 2(0) = 4 \Rightarrow x = 4 \)[/tex]
- Coordinate: [tex]\( (4, 0) \)[/tex]
3. Intersection of [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Coordinate: [tex]\( (0, 0) \)[/tex]
### Step 6: Conclusion
- The solution region is bounded (as it is restricted by the lines and does not extend infinitely).
- The coordinates of the corner points are:
[tex]\[ (0, 0), (0, 2), (4, 0) \][/tex]
So, to summarize:
- The solution region is: Bounded
- The coordinates of each corner point: [tex]\((0, 0)\)[/tex], [tex]\((0, 2)\)[/tex], [tex]\((4, 0)\)[/tex]
### Step 1: Understanding the Inequalities
We have the following system of inequalities:
1. [tex]\( x + 2 y \leq 4 \)[/tex]
2. [tex]\( x \geq 0 \)[/tex]
3. [tex]\( y \geq 0 \)[/tex]
### Step 2: Plotting the Boundary Lines
First, consider the equality for each inequality:
1. [tex]\( x + 2 y = 4 \)[/tex]
2. [tex]\( x = 0 \)[/tex]
3. [tex]\( y = 0 \)[/tex]
- [tex]\( x + 2 y = 4 \)[/tex]:
This is a straight line. We can find the intercepts:
- For [tex]\( x \)[/tex]-intercept ([tex]\(y = 0\)[/tex]):
[tex]\( x + 2(0) = 4 \Rightarrow x = 4 \)[/tex]
- For [tex]\( y \)[/tex]-intercept ([tex]\(x = 0\)[/tex]):
[tex]\( 0 + 2 y = 4 \Rightarrow 2 y = 4 \Rightarrow y = 2 \)[/tex]
So, the line passes through points [tex]\( (4, 0) \)[/tex] and [tex]\( (0, 2) \)[/tex].
- [tex]\( x = 0 \)[/tex]:
This is the [tex]\( y \)[/tex]-axis.
- [tex]\( y = 0 \)[/tex]:
This is the [tex]\( x \)[/tex]-axis.
### Step 3: Shading the Regions
- For [tex]\( x + 2 y \leq 4 \)[/tex]: We shade the region below the line [tex]\( x + 2 y = 4 \)[/tex].
- For [tex]\( x \geq 0 \)[/tex]: We shade the region to the right of the [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]).
- For [tex]\( y \geq 0 \)[/tex]: We shade the region above the [tex]\( x \)[/tex]-axis ([tex]\( y = 0 \)[/tex]).
### Step 4: Determining the Solution Region
The solution region is the intersection of all these shaded regions. The feasible region is bounded by the lines [tex]\( x + 2 y = 4 \)[/tex], [tex]\( x = 0 \)[/tex], and [tex]\( y = 0 \)[/tex].
### Step 5: Identifying the Corner Points
The corner points of the feasible solution region (where lines intersect) can be calculated:
1. Intersection of [tex]\( x + 2 y = 4 \)[/tex] and [tex]\( x = 0 \)[/tex]:
- Set [tex]\( x = 0 \)[/tex] in [tex]\( x + 2 y = 4 \)[/tex]:
[tex]\( 0 + 2 y = 4 \Rightarrow y = 2 \)[/tex]
- Coordinate: [tex]\( (0, 2) \)[/tex]
2. Intersection of [tex]\( x + 2 y = 4 \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex] in [tex]\( x + 2 y = 4 \)[/tex]:
[tex]\( x + 2(0) = 4 \Rightarrow x = 4 \)[/tex]
- Coordinate: [tex]\( (4, 0) \)[/tex]
3. Intersection of [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Coordinate: [tex]\( (0, 0) \)[/tex]
### Step 6: Conclusion
- The solution region is bounded (as it is restricted by the lines and does not extend infinitely).
- The coordinates of the corner points are:
[tex]\[ (0, 0), (0, 2), (4, 0) \][/tex]
So, to summarize:
- The solution region is: Bounded
- The coordinates of each corner point: [tex]\((0, 0)\)[/tex], [tex]\((0, 2)\)[/tex], [tex]\((4, 0)\)[/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
