https://file.aiquickdraw.com/mathbot/file/1728769993751pdcucdvf.jpg

The system of inequalities provided in the image is:

\[
\begin{cases}
y \leq 2x - 5 \\
y \geq 5x - 3
\end{cases}
\]

We need to check which of the given coordinates satisfy these inequalities. The coordinates are:

1. \( \left(-2, \frac{31}{3} \right) \)
2. \( \left(-3, \frac{40}{3} \right) \)
3. \( (-4, -9) \)

### Step-by-Step Process to Check the Solutions

#### 1. For \( \left( -2, \frac{31}{3} \right) \):

- **First inequality**: \( y \leq 2x - 5 \)
  
  Substituting \( x = -2 \):
  \[
  y \leq 2(-2) - 5 \implies y \leq -4 - 5 \implies y \leq -9
  \]
  
  For \( y = \frac{31}{3} \approx 10.33 \), clearly \( 10.33 \not\leq -9 \), so this point does **not** satisfy the first inequality.

- **Second inequality**: Since the point fails the first inequality, we don't need to check the second.

Thus, \( \left( -2, \frac{31}{3} \right) \) is **not a solution**.

#### 2. For \( \left( -3, \frac{40}{3} \right) \):

- **First inequality**: \( y \leq 2x - 5 \)

  Substituting \( x = -3 \):
  \[
  y \leq 2(-3) - 5 \implies y \leq -6 - 5 \implies y \leq -11
  \]

  For \( y = \frac{40}{3} \approx 13.33 \), clearly \( 13.33 \not\leq -11 \), so this point does **not** satisfy the first inequality.

Thus, \( \left( -3, \frac{40}{3} \right) \) is **not a solution**.

#### 3. For \( (-4, -9) \):

- **First inequality**: \( y \leq 2x - 5 \)

  Substituting \( x = -4 \):
  \[
  y \leq 2(-4) - 5 \implies y \leq -8 - 5 \implies y \leq -13
  \]

  Since \( y = -9 \), and \( -9 \leq -13 \) is **false**, this point does **not** satisfy the first inequality.

Thus, \( (-4, -9) \) is also **not a solution**.

### Conclusion:
None of the given points satisfies the system of inequalities.

Would you like further details, or any clarifications?

Here are 5 related questions to extend your understanding:
1. How can we graphically represent a system of inequalities?
2. What is the difference between strict and non-strict inequalities in systems of equations?
3. How do you find the solution region for a system of linear inequalities?
4. Can we apply substitution to solve more complex systems of inequalities?
5. How do constraints in inequalities affect real-world optimization problems?

**Tip**: Always check both inequalities when solving a system, even if a point fails the first one!

Which of the following coordinates are solutions to the system of linear inequalities: y ≤ 2x - 5, y ≥ 5x - 3?

Explore how to determine solutions for a system of linear inequalities y ≤ 2x - 5 and y ≥ 5x - 3. This problem involves checking which coordinates satisfy the inequalities, suitable for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation