Let's start by solving the given system of inequalities:
1. [tex]\( x - y > -13 \)[/tex]
2. [tex]\( 2x + y > 1 \)[/tex]
We can analyze and graph these inequalities to find the solution region.
### Step-by-Step Solution
#### Inequality 1: [tex]\( x - y > -13 \)[/tex]
First, we can rewrite the inequality in the form of an equation [tex]\( x - y = -13 \)[/tex] to better understand the boundary.
[tex]\[ x - y = -13 \][/tex]
[tex]\[ y = x + 13 \][/tex]
This is a line with a slope of 1 and a y-intercept of 13. However, since the inequality is [tex]\( x - y > -13 \)[/tex], we are interested in the region above this line.
#### Inequality 2: [tex]\( 2x + y > 1 \)[/tex]
Similarly, rewrite this inequality as an equation:
[tex]\[ 2x + y = 1 \][/tex]
[tex]\[ y = -2x + 1 \][/tex]
This line has a slope of -2 and a y-intercept of 1. The inequality [tex]\( 2x + y > 1 \)[/tex] indicates we are interested in the region above this line.
### Graphical Representation
Plot the two lines on the same coordinate plane.
1. Line 1 ([tex]\( y = x + 13 \)[/tex]):
- Slope: 1
- Y-intercept: 13
2. Line 2 ([tex]\( y = -2x + 1 \)[/tex]):
- Slope: -2
- Y-intercept: 1
### Boundary Intersection Point
To find the intersection point of the lines [tex]\( y = x + 13 \)[/tex] and [tex]\( y = -2x + 1 \)[/tex]:
[tex]\[ x + 13 = -2x + 1 \][/tex]
[tex]\[ 3x = -12 \][/tex]
[tex]\[ x = -4 \][/tex]
Plug [tex]\( x = -4 \)[/tex] back into either equation. Let's use [tex]\( y = x + 13 \)[/tex]:
[tex]\[ y = -4 + 13 \][/tex]
[tex]\[ y = 9 \][/tex]
So, the lines intersect at the point [tex]\((-4, 9)\)[/tex].
### Solution Region
- For [tex]\( x - y > -13 \)[/tex] (or [tex]\( y < x + 13 \)[/tex]), we take the region below the line [tex]\( y = x + 13 \)[/tex].
- For [tex]\( 2x + y > 1 \)[/tex] (or [tex]\( y > -2x + 1 \)[/tex]), we take the region above the line [tex]\( y = -2x + 1 \)[/tex].
The solution is the region where both conditions overlap.
### Conclusion
The solution region is the area where the lines [tex]\( y = x + 13 \)[/tex] and [tex]\( y = -2x + 1 \)[/tex] intersect and satisfy the inequalities:
- Below the line [tex]\( y = x + 13 \)[/tex]
- Above the line [tex]\( y = -2x + 1 \)[/tex]
Hence, the solution to the system of inequalities [tex]\( \{ (x, y) \mid (x - y > -13) \land (2x + y > 1) \} \)[/tex] is the region above [tex]\( y = -2x + 1 \)[/tex] and below [tex]\( y = x + 13 \)[/tex].
