To solve the system of inequalities graphically, we'll consider each inequality and plot them on a set of axes.
-
First Inequality: \( y < -x + 1 \)
- The line \( y = -x + 1 \) has a y-intercept of 1 and a slope of -1.
- This means it goes down one unit for each unit it goes to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 1 \) → point (0, 1)
- When \( x = 1 \), \( y = 0 \) → point (1, 0)
- The area below this line will represent \( y < -x + 1 \).
-
Second Inequality: \( y < \frac{1}{5}x + 7 \)
- The line \( y = \frac{1}{5}x + 7 \) has a y-intercept of 7 and a slope of \( \frac{1}{5} \).
- This means it goes up one unit for every five units to the right.
- To plot the line, we can find two points:
- When \( x = 0 \), \( y = 7 \) → point (0, 7)
- When \( x = 5 \), \( y = 8 \) → point (5, 8)
- The area below this line will represent \( y < \frac{1}{5}x + 7 \).
Graphing the Inequalities
- Draw the axes (x and y).
- Plot the line for \( y = -x + 1 \) using the points (0, 1) and (1, 0). Shade the area below this line.
- Plot the line for \( y = \frac{1}{5}x + 7 \) using the points (0, 7) and (5, 8). Shade the area below this line.
Finding the Solution Set
The solution set is the region where the shaded areas of both inequalities overlap. This region represents the values of \( x \) and \( y \) that satisfy both inequalities.
Example Point in the Solution Set
To find a point in the solution set, we can look for coordinates that lie in the overlapping shaded area. For example, the point (0, 0) can be checked:
- For the first inequality:
- \( 0 < -0 + 1 \) (True)
- For the second inequality:
- \( 0 < \frac{1}{5}(0) + 7 \) (True)
Thus, the point (0, 0) is in the solution set.
Conclusion
The coordinates of a point in the solution set are: (0, 0).