Math Problem Statement

Graph the system of inequalities: 1. 12x + 10y ≤ 120 2. 9x + 13y ≤ 117 3. x ≥ 0 4. y ≥ 0

Solution

The image shows a system of linear inequalities that needs to be graphed on a coordinate plane. The inequalities are as follows:

12x + 10y \leq 120 \\ 9x + 13y \leq 117 \\ x \geq 0 \\ y \geq 0 \end{cases}$$ ### Let's break down the steps to solve and graph the system: 1. **Rewrite the inequalities as equalities to find boundary lines:** - First inequality: $$12x + 10y = 120$$ - Second inequality: $$9x + 13y = 117$$ 2. **Find the intercepts for each line** to graph them. - **For $$12x + 10y = 120$$:** - $$x$$-intercept: Set $$y = 0$$, then $$12x = 120 \implies x = 10$$. - $$y$$-intercept: Set $$x = 0$$, then $$10y = 120 \implies y = 12$$. - **For $$9x + 13y = 117$$:** - $$x$$-intercept: Set $$y = 0$$, then $$9x = 117 \implies x = 13$$. - $$y$$-intercept: Set $$x = 0$$, then $$13y = 117 \implies y = 9$$. 3. **Graph each boundary line:** - Plot the points $$(10, 0)$$ and $$(0, 12)$$ for the first line, and connect them. - Plot the points $$(13, 0)$$ and $$(0, 9)$$ for the second line, and connect them. 4. **Determine the solution region:** - For $$12x + 10y \leq 120$$, shade the region below or on the line. - For $$9x + 13y \leq 117$$, shade the region below or on the line. - The constraints $$x \geq 0$$ and $$y \geq 0$$ indicate that the region of interest lies in the first quadrant. 5. **Find the feasible region**, which is the intersection of the shaded regions from both inequalities in the first quadrant. Would you like me to provide a detailed graph of this solution? Let me know if you have any questions or need further clarifications! ### Follow-up Questions: 1. What is the significance of the intercepts in graphing linear inequalities? 2. How do you identify the solution region of a system of inequalities? 3. How would you approach solving the system if it had three inequalities instead of two? 4. Why is it important that $$x \geq 0$$ and $$y \geq 0$$ in this problem? 5. What happens if the inequalities were strict (e.g., $$<$$ instead of $$\leq$$)? ### Tip: When solving linear inequalities, always test a point (like the origin) to check which side of the boundary line should be shaded.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Graphing Inequalities
System of Inequalities
Intercepts

Formulas

12x + 10y = 120
9x + 13y = 117

Theorems

Linear Inequalities Theorem
Intercepts Method

Suitable Grade Level

Grades 9-11