2x+y<80,x+y<50,x>0,y>0
To solve the system of inequalities 2x + y < 80, x + y < 50, x > 0, and y > 0, we can use graphical representation or substitution method.
Let's start by graphing each inequality:
1. 2x + y < 80:
To graph this inequality, we need to plot the line 2x + y = 80 and shade the region below the line (since it's a "less than" inequality).
First, rewrite the equation in slope-intercept form:
y = -2x + 80
Now plot the line by choosing two points:
Let x = 0, then y = 80.
Let x = 40, then y = 0.
Connect the two points and shade the region below the line.
2. x + y < 50:
To graph this inequality, we need to plot the line x + y = 50 and shade the region below the line.
Rewrite the equation in slope-intercept form:
y = -x + 50
Plot the line by choosing two points:
Let x = 0, then y = 50.
Let x = 50, then y = 0.
Connect the two points and shade the region below the line.
3. x > 0:
For this inequality, we only need to draw a vertical line at x = 0 and shade the region to the right of the line.
4. y > 0:
Similarly, draw a horizontal line at y = 0 and shade the region above the line.
Now, to find the region that satisfies all four inequalities, look for the shaded region where they all overlap.
The solution set for this system of inequalities is the region that satisfies all conditions: the shaded region where all four regions intersect. The specific coordinates within that region represent the values of x and y that satisfy the system.