Question

graph the solution of the system of linear inequalities.
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$$\begin{cases}x\\geq 5y \\\\ x + 2y\\geq 10\\end{cases}$$

use the graphing tool to graph the solution.
click to enlarge graph

Explanation:

Step1: Analyze \( x \geq 5y \)

Rewrite \( x \geq 5y \) as \( y \leq \frac{1}{5}x \). This is a linear inequality with a boundary line \( y = \frac{1}{5}x \) (solid line, since \( \geq \) includes equality) and the region below the line (since \( y \leq \) the line) is part of the solution.

Step2: Analyze \( x + 2y \geq 10 \)

Rewrite \( x + 2y \geq 10 \) as \( 2y \geq -x + 10 \), then \( y \geq -\frac{1}{2}x + 5 \). The boundary line is \( y = -\frac{1}{2}x + 5 \) (solid line, as \( \geq \) includes equality) and the region above the line (since \( y \geq \) the line) is part of the solution.

Step3: Find Intersection of Regions

The solution to the system is the intersection of the regions defined by \( y \leq \frac{1}{5}x \) and \( y \geq -\frac{1}{2}x + 5 \), along with the boundary lines (since both inequalities are "greater than or equal" or "less than or equal", the lines are included). To graph, plot both lines: \( y = \frac{1}{5}x \) (a line with slope \( \frac{1}{5} \), y-intercept 0) and \( y = -\frac{1}{2}x + 5 \) (slope \( -\frac{1}{2} \), y-intercept 5). Shade the area that is below \( y = \frac{1}{5}x \) and above \( y = -\frac{1}{2}x + 5 \), including the lines.

Answer:

To graph the solution:

1. Plot the line \( y = \frac{1}{5}x \) (solid, slope \( \frac{1}{5} \), y-intercept 0) and shade below it.
2. Plot the line \( y = -\frac{1}{2}x + 5 \) (solid, slope \( -\frac{1}{2} \), y-intercept 5) and shade above it.
3. The solution is the overlapping shaded region (including both lines).

(Note: The graphing tool can be used to visualize this by plotting the two lines and identifying the intersection of the shaded regions.)