Graph the lines y ≥ −x+7 and y > 2x−5. Shaded overlapping region is the solution. A point in the solution set is (2, 1).
To solve the system of inequalities y ≥ −x+7 and y > 2x−5 graphically, follow these steps:
Start by graphing the boundary lines for each inequality. For y≥−x+7, the boundary line is y = −x+7. For y > 2x−5, the boundary line is
y = 2x−5.
Let's graph these lines on the set of axes:
y = −x+7
To graph this line, plot the y-intercept at (0, 7) and then use the slope (-1) to find another point. The slope tells you to go down 1 unit and to the right 1 unit.
y = 2x−5
For this line, plot the y-intercept at (0, -5) and use the slope (2) to find another point. The slope tells you to go up 2 units and to the right 1 unit.
Now, you need to determine which side of each line to shade. For y ≥ −x+7, shade the region above the line because it includes all points where y is greater than or equal to the expression on the right side of the inequality.
For y > 2x−5, shade the region above the line because it includes all points where y is greater than the expression on the right side of the inequality.
The solution to the system of inequalities is the overlapping shaded region. It's the region where both shaded areas overlap.
Pick any point within the overlapping shaded region as a solution. For simplicity, you can choose the coordinates of the point where the shaded regions overlap.
State the coordinates of the chosen point as part of the solution set.
Now, let's illustrate this process on the graph. I'll describe the graph visually:
Graph the line
y = −x+7.
Plot the y-intercept at (0, 7).
Use the slope (-1) to find another point by going down 1 unit and right 1 unit.
Graph the line
y = 2x−5.
Plot the y-intercept at (0, -5).
Use the slope (2) to find another point by going up 2 units and right 1 unit.
Shade the region above both lines.
The solution is the overlapping shaded region.