Which term best describes the solution of the situation represented by the system of inequalities? (Assume that x >= 0 and y >= 0.)
3x + 4y <= 8
2x + 4y <= 6
Answer choices given are:
a. alternate optimal solutions
b. one optimal solution
c. unbounded
d. infeasible
I have no clue how to do this and what they are asking for. (I can't seem to find any video covering this in Khan Academy either.)
Consider each line as if it were an equation. For example,
3x+4y = 8
That describes a line in the x-y plane. But, you have an inequality. So, the solution is the entire half-plane below the line, where 3x+4y <= 8
Same for the other inequality.
So, the solution set for the system is the intersection of those two regions: the entire plane which is below both lines.
See the solution at
http://www.wolframalpha.com/input/?i=solve+3x+%2B+4y+%3C%3D+8%2C+2x+%2B+4y+%3C%3D+6
To determine the solution of the system of inequalities, you need to find the region of the coordinate plane that satisfies both inequalities simultaneously. Here's how you can approach this:
1. Start by graphing the boundary lines of each inequality.
For the first inequality, 3x + 4y <= 8, you can choose a few values for x and solve for y to find the corresponding points on the line. For example, when x = 0, you have 4y <= 8, which gives y <= 2. So, one point on this line is (0, 2). Repeat this process for a few more values of x to get additional points, and then connect them to plot the boundary line.
Repeat the same steps for the second inequality, 2x + 4y <= 6, and plot its boundary line.
2. Now, shade the region that satisfies both inequalities.
To determine the region that satisfies both inequalities, you need to shade the area where the shaded regions of each inequality overlap. This region will contain all the points that satisfy both inequalities simultaneously.
3. Analyze the shaded region and determine the answer.
Once you have shaded the region, you need to analyze its characteristics based on the answer choices.
- If the shaded region is a finite area with a clear minimum or maximum point, it indicates a single optimal solution. In this case, choose option b. "one optimal solution."
- If the shaded region is a finite area without a clear minimum or maximum point, it suggests multiple solutions that are equally optimal. In this case, choose option a. "alternate optimal solutions."
- If the shaded region extends indefinitely in one or more directions, it suggests an unbounded solution. In this case, choose option c. "unbounded."
- If the shaded region does not exist and there is no overlap between the two boundary lines, it suggests no possible solution. In this case, choose option d. "infeasible."
By following these steps and analyzing the characteristics of the shaded region, you can determine the best term that describes the solution of the given system of inequalities.