x+y≤23x+y≥3x+4y≤3x≥0y≥0
list the verticies

To list the vertices for the system of inequalities, let's first rewrite the inequalities clearly:

1. \( x + y \leq 23 \)
2. \( x + y \geq 3 \)
3. \( x + 4y \leq 3 \)
4. \( x \geq 0 \)  (x must be non-negative)
5. \( y \geq 0 \)  (y must be non-negative)

These inequalities define a region in the coordinate plane. To find the vertices of the region, we need to solve the system by finding the points where the boundary lines intersect.

**Step 1: Identify the lines**
- \( x + y = 23 \)
- \( x + y = 3 \)
- \( x + 4y = 3 \)
- \( x = 0 \) (y-axis)
- \( y = 0 \) (x-axis)

**Step 2: Find the intersection points (vertices)**

1. **Intersection of \( x + y = 23 \) and \( x + y = 3 \)**

   Subtract the second equation from the first:
   \[
   (x + y) - (x + y) = 23 - 3 \implies 0 = 20
   \]
   These lines are parallel and do not intersect.

2. **Intersection of \( x + y = 23 \) and \( x + 4y = 3 \)**

   Solve the system:
   - \( x + y = 23 \)  → \( x = 23 - y \)
   - Substituting into \( x + 4y = 3 \):
     \[
     (23 - y) + 4y = 3 \implies 23 + 3y = 3 \implies 3y = -20 \implies y = -\frac{20}{3}
     \]
     Since \( y \geq 0 \), this solution is not valid.

3. **Intersection of \( x + y = 3 \) and \( x + 4y = 3 \)**

   Solve the system:
   - \( x + y = 3 \)  → \( x = 3 - y \)
   - Substituting into \( x + 4y = 3 \):
     \[
     (3 - y) + 4y = 3 \implies 3 + 3y = 3 \implies 3y = 0 \implies y = 0
     \]
     Substituting \( y = 0 \) into \( x + y = 3 \), we get \( x = 3 \).
     Thus, one vertex is \( (3, 0) \).

4. **Intersection of \( x + 4y = 3 \) and \( x = 0 \)**

   Substituting \( x = 0 \) into \( x + 4y = 3 \):
   \[
   0 + 4y = 3 \implies y = \frac{3}{4}
   \]
   Thus, another vertex is \( (0, \frac{3}{4}) \).

5. **Intersection of \( x + y = 23 \) and \( x = 0 \)**

   Substituting \( x = 0 \) into \( x + y = 23 \):
   \[
   0 + y = 23 \implies y = 23
   \]
   Thus, another vertex is \( (0, 23) \).

**Step 3: List the vertices**

The vertices of the region are:
1. \( (3, 0) \)
2. \( (0, \frac{3}{4}) \)
3. \( (0, 23) \)

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. How do we determine if a system of inequalities forms a bounded region?
2. What is the significance of vertices in optimization problems (e.g., linear programming)?
3. How do parallel lines affect the solution to a system of inequalities?
4. What happens if two inequalities intersect outside the feasible region?
5. Can the number of vertices in a system of inequalities vary, and how?

**Tip**: Always graph the inequalities to visualize the feasible region and make identifying vertices easier.

The query submitted by the user: x + y ≤ 23, x + y ≥ 3, x + 4y ≤ 3, x ≥ 0, y ≥ 0. List the vertices.

Learn how to solve a system of linear inequalities, including x + y ≤ 23 and x + 4y ≤ 3, by identifying and listing the vertices of the feasible region. This step-by-step guide explores intersection points and graphing techniques for high school students.

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