3x+5y≤30 x≤9 y≤3 x≥0 y≥0
Graphing the inequalities gives us the following region:
The shaded region in the graph represents all possible values of x and y that satisfy the given inequalities.
3x+5y≤30 x≤9 y≤3 x≥0 y≥0
Here is the graph of the given system of inequalities:
The shaded region represents all values of x and y that satisfy the given inequalities.
To solve this system of inequalities step-by-step, let's break it down and find the overlapping region that satisfies all the given conditions.
Step 1: Plot the inequality 3x + 5y ≤ 30 on a coordinate system:
To plot this inequality, we need to rewrite it in the form of y = mx + b:
3x + 5y ≤ 30
5y ≤ -3x + 30
y ≤ (-3/5)x + 6
Now, graph the line y = (-3/5)x + 6, but include the boundary line as a dashed line since the inequality has ≤:
- Plot the y-intercept at (0, 6).
- Use the slope (-3/5) to plot additional points, such as (5, 3) and (10, 0).
- Draw a dashed line through these points extending it indefinitely in both directions.
Step 2: Plot the inequality x ≤ 9:
Draw a vertical line at x = 9.
Step 3: Plot the inequality y ≤ 3:
Draw a horizontal line at y = 3.
Step 4: Plot the inequalities x ≥ 0 and y ≥ 0:
Draw two lines, one vertical (x = 0) and one horizontal (y = 0), to create the positive quadrant of the coordinate system.
Step 5: Identify the overlapping region:
The overlapping region is the shaded area where all the conditions are satisfied. In this case, it is the region below the line y = (-3/5)x + 6, to the left of x = 9, below y = 3, and in the positive quadrant (to the right and above the lines x = 0 and y = 0).
To summarize in words, the overlapping region that satisfies all the given inequalities is:
- x is between 0 and 9 (inclusive),
- y is between 0 and 3 (inclusive),
- and the points in the region are below the line y = (-3/5)x + 6.
I hope this helps! Let me know if you have any further questions.
The given set of inequalities can be represented as follows:
1) 3x + 5y ≤ 30
2) x ≤ 9
3) y ≤ 3
4) x ≥ 0
5) y ≥ 0
To determine the solution to this system of inequalities, we need to find the feasible region where all the inequalities are satisfied simultaneously.
1) Start by graphing the equations on a coordinate plane.
To graph the first inequality, 3x + 5y ≤ 30, we can rewrite it in slope-intercept form: y ≤ (-3/5)x + 6. Plot the line y = (-3/5)x + 6 (which has a y-intercept of 6 and a slope of -3/5), and shade the area below the line to represent the inequality.
2) Now graph the remaining four inequalities.
- The inequality x ≤ 9 represents all the points on or to the left of the vertical line passing through x = 9. Shade the left side of the line.
- The inequality y ≤ 3 represents all the points on or below the horizontal line passing through y = 3. Shade the area below the line.
- The inequality x ≥ 0 represents all the points on or to the right of the y-axis. Shade the right side of the y-axis.
- The inequality y ≥ 0 represents all the points above the x-axis. Shade the area above the x-axis.
3) The feasible region is the overlapping region of all the shaded regions. It is the region that satisfies all the given inequalities.
In this case, the feasible region is the triangle-shaped region below the line y = (-3/5)x + 6, to the left of x = 9, and below y = 3. The region is bounded by the x and y axes on the lower sides.
4) To find the solution, we need to list the coordinates of the vertices of the feasible region. In this case, the vertices are:
(0, 0)
(0, 3)
(9, 3)
Therefore, the solution to the inequalities is the set of all points within or on the triangle with vertices (0, 0), (0, 3), and (9, 3), including the boundaries.