Answer:
Region C of the graph will contains the solution to the given system
Step-by-step explanation:
Consider the given system of equation
[tex]y\le\frac{1}{4}x+3[/tex]
and [tex]y\ge-x+5[/tex]
We have to choose the region of the graph that contains the solution to the given system.
Since, to determine the region choose a test point in each region and then then check the values of inequality at that point and for the test point that satisfies both the inequality will contains the solution to the given system.
On region A)
Let (0, 4) be the test point that lies in region A
Then put the value of x = 0 and y= 4 in given system,
we have,
[tex]4\le\frac{1}{4}(0)+3 \Rightarrow 4\le 3[/tex] (false)
and [tex]4\ge(0)+5 \Rightarrow 4\ge 5[/tex] (false)
On region B)
Let (0, 8) be the test point that lies in region B
Then put the value of x = 0 and y= 8 in given system,
we have,
[tex]8\le\frac{1}{4}(0)+3 \Rightarrow 8\le 3[/tex] (false)
and [tex]8\ge(0)+5 \Rightarrow 8\ge 5[/tex] (true)
On region C)
Let (8,0) be the test point that lies in region C
Then put the value of x = 8 and y= 0 in given system,
we have,
[tex]0\le\frac{1}{4}(8)+3 \Rightarrow 0\le 5[/tex] (true)
and [tex]0\ge(-8)+5 \Rightarrow 0\ge -3[/tex] (true)
On region D)
Let (0,0) be the test point that lies in region D
Then put the value of x = 0 and y= 0 in given system,
we have,
[tex]0\le\frac{1}{4}(0)+3 \Rightarrow 0\le 3[/tex] (false)
and [tex]0\ge(0)+5 \Rightarrow 0\ge 5[/tex] (false)
Thus, only (8,0) be the test point that lies in region C satisfies both inequality.
Thus, region C of the graph will contains the solution to the given system
