Answer :
To solve this system of inequalities and determine the green region representing the answer, let’s analyze each inequality step by step.
1. Consider the first inequality:
[tex]\[ x + 2y \leq 4 \][/tex]
To find the boundary line for this inequality, we replace the inequality sign with an equal sign:
[tex]\[ x + 2y = 4 \][/tex]
We can express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] and solve for one variable. Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 4 - 2y \][/tex]
This is the equation of a line, and it will help us determine one boundary of our solution region.
2. Consider the second inequality:
[tex]\[ 3x - y > 2 \][/tex]
Similarly, replace the inequality sign with an equal sign to find the boundary line:
[tex]\[ 3x - y = 2 \][/tex]
Solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[ 3x = y + 2 \][/tex]
[tex]\[ x = \frac{y + 2}{3} \][/tex]
This is another boundary line.
Next, we need the potential boundary points, which we obtain by solving the above equations simultaneously:
[tex]\[ \begin{cases} x + 2y = 4 \\ 3x - y = 2 \end{cases} \][/tex]
Solving these equations together:
1. Start with:
[tex]\[ x = 4 - 2y \][/tex]
Substitute this [tex]\(x\)[/tex] in the second equation:
[tex]\[ 3(4 - 2y) - y = 2 \][/tex]
[tex]\[ 12 - 6y - y = 2 \][/tex]
[tex]\[ 12 - 7y = 2 \][/tex]
[tex]\[ -7y = 2 - 12 \][/tex]
[tex]\[ -7y = -10 \][/tex]
[tex]\[ y = \frac{10}{7} \][/tex]
2. Substitute [tex]\(y = \frac{10}{7}\)[/tex] back into [tex]\(x = 4 - 2y\)[/tex]:
[tex]\[ x = 4 - 2 \left(\frac{10}{7}\right) \][/tex]
[tex]\[ x = 4 - \frac{20}{7} \][/tex]
[tex]\[ x = \frac{28}{7} - \frac{20}{7} \][/tex]
[tex]\[ x = \frac{8}{7} \][/tex]
Thus, the solution to the system when treated as equalities yields the boundary points:
[tex]\[ (x, y) = \left(\frac{8}{7}, \frac{10}{7}\right) \][/tex]
These boundary points define the lines, and we need to check the inequalities to find the feasible region. For the green region:
- The line [tex]\(x + 2y = 4\)[/tex] accounts for the boundary, and the solution is below or on this line.
- The line [tex]\(3x - y = 2\)[/tex] provides another boundary, but the feasible region is above this line.
To determine the solution region, you can graph the inequalities and shade the regions satisfying both conditions. The intersection of these two shaded areas represents the solution to the system of inequalities.
Therefore, the green region that represents the answer region is bounded by the lines [tex]\(x + 2y \leq 4\)[/tex] and [tex]\(3x - y > 2\)[/tex].
1. Consider the first inequality:
[tex]\[ x + 2y \leq 4 \][/tex]
To find the boundary line for this inequality, we replace the inequality sign with an equal sign:
[tex]\[ x + 2y = 4 \][/tex]
We can express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] and solve for one variable. Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 4 - 2y \][/tex]
This is the equation of a line, and it will help us determine one boundary of our solution region.
2. Consider the second inequality:
[tex]\[ 3x - y > 2 \][/tex]
Similarly, replace the inequality sign with an equal sign to find the boundary line:
[tex]\[ 3x - y = 2 \][/tex]
Solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[ 3x = y + 2 \][/tex]
[tex]\[ x = \frac{y + 2}{3} \][/tex]
This is another boundary line.
Next, we need the potential boundary points, which we obtain by solving the above equations simultaneously:
[tex]\[ \begin{cases} x + 2y = 4 \\ 3x - y = 2 \end{cases} \][/tex]
Solving these equations together:
1. Start with:
[tex]\[ x = 4 - 2y \][/tex]
Substitute this [tex]\(x\)[/tex] in the second equation:
[tex]\[ 3(4 - 2y) - y = 2 \][/tex]
[tex]\[ 12 - 6y - y = 2 \][/tex]
[tex]\[ 12 - 7y = 2 \][/tex]
[tex]\[ -7y = 2 - 12 \][/tex]
[tex]\[ -7y = -10 \][/tex]
[tex]\[ y = \frac{10}{7} \][/tex]
2. Substitute [tex]\(y = \frac{10}{7}\)[/tex] back into [tex]\(x = 4 - 2y\)[/tex]:
[tex]\[ x = 4 - 2 \left(\frac{10}{7}\right) \][/tex]
[tex]\[ x = 4 - \frac{20}{7} \][/tex]
[tex]\[ x = \frac{28}{7} - \frac{20}{7} \][/tex]
[tex]\[ x = \frac{8}{7} \][/tex]
Thus, the solution to the system when treated as equalities yields the boundary points:
[tex]\[ (x, y) = \left(\frac{8}{7}, \frac{10}{7}\right) \][/tex]
These boundary points define the lines, and we need to check the inequalities to find the feasible region. For the green region:
- The line [tex]\(x + 2y = 4\)[/tex] accounts for the boundary, and the solution is below or on this line.
- The line [tex]\(3x - y = 2\)[/tex] provides another boundary, but the feasible region is above this line.
To determine the solution region, you can graph the inequalities and shade the regions satisfying both conditions. The intersection of these two shaded areas represents the solution to the system of inequalities.
Therefore, the green region that represents the answer region is bounded by the lines [tex]\(x + 2y \leq 4\)[/tex] and [tex]\(3x - y > 2\)[/tex].
