Math Problem Statement

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

Solution

The system of inequalities provided in the image is:

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!

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Coordinate Geometry
Systems of Inequalities

Formulas

y ≤ 2x - 5
y ≥ 5x - 3

Theorems

Graphing linear inequalities

Suitable Grade Level

Grades 9-11

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